The Weyl-Heisenberg symmetries originate from translation invariances of various manifolds viewed as phase spaces, e.g. Euclidean plane, semi-discrete cylinder, torus, in the two-dimensional case, and higher-dimensional generalisations. In this review we describe, on an elementary level, how this symmetry emerges through displacement operators and standard Fourier analysis, and how their unitary representations are used both in Signal Analysis (time-frequency techniques, Gabor transform) and in quantum formalism (covariant integral quantizations and semi-classical portraits). An example of application of the formalism to the Majorana stellar constellation in the plane is presented.

I consider the long-standing issue of the hermicity of the Dirac equation in curved spacetime metrics. Instead of imposing hermiticity by adding ad hoc terms, I renormalize the field by a scaling function, which is related to the determinant of the metric, and then regularize the renormalized field on a discrete lattice. I found that, for time-independent and diagonal metrics such as the Rindler, de~Sitter, and anti-de~Sitter metrics, the Dirac equation returns a hermitian or pseudohermitian ($\mathcal{PT}$-symmetric) Hamiltonian when properly regularized on the lattice. Notably, the $\mathcal{PT}$-symmetry is unbroken in the pseudohermitian cases, assuring a real energy spectrum with unitary time evolution. Conversely, considering a more general class of time-dependent metrics, which includes the Weyl metric, the Dirac equation returns a nonhermitian Hamiltonian with nonunitary time evolution. Arguably, this nonhermicity is physical, with the time dependence of the metric corresponding to local nonhermitian processes on the lattice and nonunitary growth or decay of the time evolution of the field. This suggests a duality between nonhermitian gain and loss phenomena and spacetime contractions and expansions. This metric-induced nonhermiticity unveils an unexpected connection between spacetime metric and nonhermitian phases of matter.

As new advancements in the field of quantum computing lead to the development of increasingly complex programs, approaches to validate and debug these programs are becoming more important. To this end, methods employed in classical debugging, such as assertions for testing specific properties of a program's state, have been adapted for quantum programs. However, to efficiently debug quantum programs, it is key to properly place these assertions. This usually requires a deep understanding of the program's underlying mathematical properties, constituting a time-consuming manual task for developers. To address this problem, this work proposes methods for automatically refining assertions in quantum programs by moving them to more favorable positions in the program or by placing new assertions that help to further narrow down potential error locations. This allows developers to take advantage of rich and expressive assertions that greatly improve the debugging experience without requiring them to place these assertions manually in an otherwise tedious manner. An open-source implementation of the proposed methods is available at https://github.com/cdatum/mqt-debugger.

Quantum error correction is essential for bridging the gap between the error rates of physical devices and the extremely low logical error rates required for quantum algorithms. Recent error-correction demonstrations on superconducting processors have focused primarily on the surface code, which offers a high error threshold but poses limitations for logical operations. In contrast, the color code enables much more efficient logic, although it requires more complex stabilizer measurements and decoding techniques. Measuring these stabilizers in planar architectures such as superconducting qubits is challenging, and so far, realizations of color codes have not addressed performance scaling with code size on any platform. Here, we present a comprehensive demonstration of the color code on a superconducting processor, achieving logical error suppression and performing logical operations. Scaling the code distance from three to five suppresses logical errors by a factor of $\Lambda_{3/5}$ = 1.56(4). Simulations indicate this performance is below the threshold of the color code, and furthermore that the color code may be more efficient than the surface code with modest device improvements. Using logical randomized benchmarking, we find that transversal Clifford gates add an error of only 0.0027(3), which is substantially less than the error of an idling error correction cycle. We inject magic states, a key resource for universal computation, achieving fidelities exceeding 99% with post-selection (retaining about 75% of the data). Finally, we successfully teleport logical states between distance-three color codes using lattice surgery, with teleported state fidelities between 86.5(1)% and 90.7(1)%. This work establishes the color code as a compelling research direction to realize fault-tolerant quantum computation on superconducting processors in the near future.

We investigate ensembles of Matrix Product States (MPSs) generated by quantum circuit evolution followed by projection onto MPSs with a fixed bond dimension $\chi$. Specifically, we consider ensembles produced by: (i) random sequential unitary circuits, (ii) random brickwork unitary circuits, and (iii) circuits involving both unitaries and projective measurements. In all cases, we characterize the spectra of the MPS transfer matrix and show that, for the first two cases in the thermodynamic limit, they exhibit a finite universal value of the spectral gap in the limit of large $\chi$, albeit with different spectral densities. We show that a finite gap in this limit does not imply a finite correlation length, as the mutual information between two large subsystems increases with $\chi$ in a manner determined by the entire shape of the spectral density. The latter differs for different types of circuits, indicating that these ensembles of MPS retain relevant physical information about the underlying microscopic dynamics. In particular, in the presence of monitoring, we demonstrate the existence of a measurement-induced entanglement transition (MIPT) in MPS ensembles, with the averaged dimension of the transfer matrix's null space serving as the effective order parameter.

We explore the dissipative phase transition of the two-photon Dicke model, a topic that has garnered significant attention recently. Our analysis reveals that while single-photon loss does not stabilize the intrinsic instability in the model, the inclusion of two-photon loss restores stability, leading to the emergence of superradiant states which coexist with the normal vacuum states. Using a second-order cumulant expansion for the photons, we derive an analytical description of the system in the thermodynamic limit which agrees well with the exact calculation results. Additionally, we present the Wigner function for the system, shedding light on the breaking of the Z4-symmetry inherent in the model. These findings offer valuable insights into stabilization mechanisms in open quantum systems and pave the way for exploring complex nonlinear dynamics in two-photon Dicke models.

Optical vector modes (VMs), characterized by spatially varying polarization distributions, have become essential tools across microscopy, metrology, optical trapping, nanophotonics, and optical communications. The Hong-Ou-Mandel (HOM) effect, a fundamental two-photon interference phenomenon in quantum optics, offers significant potential to extend the applications of VMs beyond the classical regime. Here, we demonstrate the simultaneous generation of all four Bell states by exploiting the HOM interference of VMs. The resulting Bell states exhibit spatially tailored distributions that are determined by the input modes. These results represent a significant step in manipulating HOM interference within structured photons, offering promising avenues for high-dimensional quantum information processing and in particular high-dimensional quantum communication, quantum sensing, and advanced photonic technologies reliant on tailored quantum states of light.

The Dicke-Ising model, one of the few paradigmatic models of matter-light interaction, exhibits a superradiant quantum phase transition above a critical coupling strength. However, in natural optical systems, its experimental validation is hindered by a "no-go theorem''. Here, we propose a digital-analog quantum simulator for this model based on an ensemble of interacting qubits coupled to a single-mode photonic resonator. We analyze the system's free energy landscape using field-theoretical methods and develop a digital-analog quantum algorithm that disentangles qubit and photon degrees of freedom through a parity-measurement protocol. This disentangling enables the emulation of a photonic Schr\"odinger cat state, which is a hallmark of the superradiant ground state in finite-size systems and can be unambiguously probed through the Wigner tomography of the resonator's field.

Nonlocal quantum games provide proof of principle that quantum resources can confer advantage at certain tasks. They also provide a compelling way to explore the computational utility of phases of matter on quantum hardware. In a recent manuscript [Hart et al., arXiv:2403.04829] we demonstrated that a toric code resource state conferred advantage at a certain nonlocal game, which remained robust to small deformations of the resource state. In this manuscript we demonstrate that this robust advantage is a generic property of resource states drawn from topological or fracton ordered phases of quantum matter. To this end, we illustrate how several other states from paradigmatic topological and fracton ordered phases can function as resources for suitably defined nonlocal games, notably the three-dimensional toric-code phase, the X-cube fracton phase, and the double-semion phase. The key in every case is to design a nonlocal game that harnesses the characteristic braiding processes of a quantum phase as a source of contextuality. We unify the strategies that take advantage of mutual statistics by relating the operators to be measured to order and disorder parameters of an underlying generalized symmetry-breaking phase transition. Finally, we massively generalize the family of games that admit perfect strategies when codewords of homological quantum error-correcting codes are used as resources.

Multipartite entangled states are fundamental resources for multi-user quantum cryptographic tasks. Despite significant advancements in generating large-scale continuous-variable (CV) cluster states, particularly the dual-rail cluster state because of its utility in measurement-based quantum computation, its application in quantum cryptography has remained largely unexplored. In this paper, we introduce a novel protocol for generating three user conference keys using a CV dual-rail cluster state. We develop the concept of a quotient graph state by applying a node coloring scheme to the infinite dual-rail graph, resulting in a six-mode pure graph state suitable for cryptographic applications. Our results demonstrate that the proposed protocol achieves performance close to that of GHZ-based protocols for quantum conference key agreement (QCKA), with GHZ states performing slightly better. However, a key advantage of our protocol lies in its ability to generate bipartite keys post-QCKA, a feature not achievable with GHZ states. Additionally, compared to a downstream access network using two-mode squeezed vacuum states, our protocol achieves superior performance in generating bipartite keys. Furthermore, we extend our analysis to the finite-size regime and consider the impact of using impure squeezed states for generating the multipartite entangled states, reflecting experimental imperfections. Our findings indicate that even with finite resources and non-ideal state preparation, the proposed protocol maintains its advantages. We also introduce a more accurate method to estimate the capacity of a protocol to generate bipartite keys in a quantum network.

Discrete-step walks describe the dynamics of particles in a lattice subject to hopping or splitting events at discrete times. Despite being of primordial interest to the physics of quantum walks, the topological properties arising from their discrete-step nature have been hardly explored. Here we report the observation of topological phases unique to discrete-step walks. We use light pulses in a double-fibre ring setup whose dynamics maps into a two-dimensional lattice subject to discrete splitting events. We show that the number of edge states is not simply described by the bulk invariants of the lattice (i.e., the Chern number and the Floquet winding number) as would be the case in static lattices and in lattices subject to smooth modulations. The number of edge states is also determined by a topological invariant associated to the discrete-step unitary operators acting at the edges of the lattice. This situation goes beyond the usual bulk-edge correspondence and allows manipulating the number of edge states without the need to go through a gap closing transition. Our work opens new perspectives for the engineering of topological modes for particles subject to quantum walks.

Nanolasers operating at low power levels are strongly affected by intrinsic quantum noise, affecting both intensity fluctuations and laser coherence. Starting from semi-classical rate equations and making a simple hypothesis for the phase of the laser field, a simple stochastic model for the laser quantum noise is suggested. The model is shown to agree quantitatively with quantum master equations for microscopic lasers with a small number of emitters and with classical Langevin equations for macroscopic systems. In contrast, neither quantum master equations nor classical Langevin equations adequately address the mesoscopic regime. The stochastic approach is used to calculate the linewidth throughout the transition to lasing, where the linewidth changes from being dominated by the particle-like nature of photons below threshold to the wave-like nature above threshold, where it is strongly influenced by index fluctuations enhancing the linewidth.

The stellar hierarchy of quantum states of light classifies the states according to the Fock-state resources that are required for their generation together with unitary Gaussian operations. States with stellar rank n can be also equivalently referred to as genuinely n-photon quantum non-Gaussian states. Here we present an efficient method for construction of general witnesses of the stellar rank. The number of parameters that need to be optimized in order to determine the witness does not depend on the stellar rank and it scales quadratically with the number of modes. We illustrate the procedure by constructing stellar rank witnesses based on pairs of Fock state probabilities and also based on pairs of fidelities with superpositions of coherent states.

A remarkable characteristic of quantum computing is the potential for reliable computation despite faulty qubits. This can be achieved through quantum error correction, which is typically implemented by repeatedly applying static syndrome checks, permitting correction of logical information. Recently, the development of time-dynamic approaches to error correction has uncovered new codes and new code implementations. In this work, we experimentally demonstrate three time-dynamic implementations of the surface code, each offering a unique solution to hardware design challenges and introducing flexibility in surface code realization. First, we embed the surface code on a hexagonal lattice, reducing the necessary couplings per qubit from four to three. Second, we walk a surface code, swapping the role of data and measure qubits each round, achieving error correction with built-in removal of accumulated non-computational errors. Finally, we realize the surface code using iSWAP gates instead of the traditional CNOT, extending the set of viable gates for error correction without additional overhead. We measure the error suppression factor when scaling from distance-3 to distance-5 codes of $\Lambda_{35,\text{hex}} = 2.15(2)$, $\Lambda_{35,\text{walk}} = 1.69(6)$, and $\Lambda_{35,\text{iSWAP}} = 1.56(2)$, achieving state-of-the-art error suppression for each. With detailed error budgeting, we explore their performance trade-offs and implications for hardware design. This work demonstrates that dynamic circuit approaches satisfy the demands for fault-tolerance and opens new alternative avenues for scalable hardware design.

Expectation value estimation is ubiquitous in quantum algorithms. The expectation value of a Hamiltonian, which is essential in various practical applications, is often estimated by measuring a large number of Pauli strings on quantum computers and performing classical post-processing. In the case of $n$-qubit molecular Hamiltonians in quantum chemistry calculations, it is necessary to evaluate $O(n^4)$ Pauli strings, requiring a large number of measurements for accurate estimation. To reduce the measurement cost, we assess an existing idea that uses two copies of an $n$-qubit quantum state of interest and coherently measures them in the Bell basis, which enables the simultaneous estimation of the absolute values of expectation values of all the $n$-qubit Pauli strings. We numerically investigate the efficiency of energy estimation for molecular Hamiltonians of up to 12 qubits. The results show that, when the target precision is no smaller than tens of milli-Hartree, this method requires fewer measurements than conventional sampling methods. This suggests that the method may be useful for many applications that rely on expectation value estimation of Hamiltonians and other observables as well when moderate precision is sufficient.

Quantum homogenization is a reservoir-based quantum state approximation protocol, which has been successfully implemented in state transformation on quantum hardware. In this work we move beyond that and propose the homogenization as a novel platform for quantum state stabilization and information protection. Using the Heisenberg exchange interactions formalism, we extend the standard quantum homogenization protocol to the dynamically-equivalent ($\mathtt{SWAP}$)$^\alpha$ formulation. We then demonstrate its applicability on available noisy intermediate-scale quantum (NISQ) processors by presenting a shallow quantum circuit implementation consisting of a sequence of $\mathtt{CNOT}$ and single-qubit gates. In light of this, we employ the Beny-Oreshkov generalization of the Knill-Laflamme (KL) conditions for near-optimal recovery channels to show that our proposed ($\mathtt{SWAP}$)$^\alpha$ quantum homogenization protocol yields a completely positive, trace preserving (CPTP) map under which the code subspace is correctable. Therefore, the protocol protects quantum information contained in a subsystem of the reservoir Hilbert space under CPTP dynamics.

A universal numerical method is developed for the investigation of magnetic neutron scattering. By applying the pseudospectral-time-domain (PSTD) algorithm to the spinor version of the Schr\"odinger equation, the evolution of the spin-state of the scattered wave can be solved in full space and time. This extra spin degree of freedom brings some unique new features absent in the numerical theory on the scalar wave scatterings [1]. Different numerical stability condition has to be re-derived due to the coupling between the different spin states. As the simplest application, the neutron scattering by the magnetic field of a uniformly magnetized sphere is studied. The PSTD predictions are compared with those from the Born-approximation. This work not only provides a systematic tool for analyzing spin-matter interactions, but also builds the forward model for testing novel neutron imaging methodologies, such as the newly developed thermal neutron Fourier-transform ghost imaging.

Machine learning has been revolutionizing our world over the last few years and is also increasingly exploited in several areas of physics, including quantum dynamics and control.The need for a framework that brings together machine learning models and quantum simulation methods has been quite high within the quantum control field, with the ultimate goal of exploiting these powerful computational methods for the efficient implementation of modern quantum technologies. The existing frameworks for quantum system simulations, such as QuTip and QuantumOptics.jl, even though they are very successful in simulating quantum dynamics, cannot be easily incorporated into the platforms used for the development of machine learning models, like for example PyTorch. The TorchQC framework introduced in the present work comes exactly to fill this gap. It is a new library written entirely in Python and based on the PyTorch deep learning library. PyTorch and other deep learning frameworks are based on tensors, a structure that is also used in quantum mechanics. This is the common ground that TorchQC utilizes to combine quantum physics simulations and deep learning models.TorchQC exploits PyTorch and its tensor mechanism to represent quantum states and operators as tensors, while it also incorporates all the tools needed to simulate quantum system dynamics. All necessary operations are internal in the PyTorch library, thus TorchQC programs can be executed in GPUs, substantially reducing the simulation time. We believe that the proposed TorchQC library has the potential to accelerate the development of deep learning models directly incorporating quantum simulations, enabling the easier integration of these powerful techniques in modern quantum technologies.

Assembling increasingly larger-scale defect-free optical tweezer-trapped atom arrays is essential for quantum computation and quantum simulations based on atoms. Here, we propose an AI-enabled, rapid, constant-time-overhead rearrangement protocol, and we experimentally assemble defect-free 2D and 3D atom arrays with up to 2024 atoms with a constant time cost of 60 ms. The AI model calculates the holograms for real-time atom rearrangement. With precise controls over both position and phase, a high-speed spatial light modulator moves all the atoms simultaneously. This protocol can be readily used to generate defect-free arrays of tens of thousands of atoms with current technologies, and become a useful toolbox for quantum error correction.

In this review article we summarize all experiments claiming quantum computational advantage to date. Our review highlights challenges, loopholes, and refutations appearing in subsequent work to provide a complete picture of the current statuses of these experiments. In addition, we also discuss theoretical computational advantage in example problems such as approximate optimization and recommendation systems. Finally, we review recent experiments in quantum error correction -- the biggest frontier to reach experimental quantum advantage in Shor's algorithm.

Here we present a derivation of the hierarchical equations of motion (HEOM) for an overdamped Lorentz-Drude environment containing an undamped oscillator (LDUO-HEOM). The new approach avoids the artifact of superfluous damping inherent in underdamped spectral densities. We show that the new HEOM is a useful model for intramolecular vibrations in condensed phase molecules.

Non-Hermitian systems exhibit a variety of unique features rooted in the presence of exceptional points (EP). The distinct topological structure in the proximity of an EP gives rise to counterintuitive behaviors absent in Hermitian systems, which emerge after encircling the EP either quasistatically or dynamically. However, experimental exploration of EP encirclement in quantum systems, particularly those involving high-order EPs, remains challenging due to the difficulty of coherently controlling more degrees of freedom. In this work, we experimentally investigate the eigenvalues braiding and state transfer arising from the encirclement of EP in a three-dimensional non-Hermitian quantum system using superconducting circuits. We characterize the second- and third-order EPs through the coalescence of eigenvalues. Then we reveal the topological structure near the EP3 by quasistatically encircling it along various paths with three independent parameters, which yields the eigenvalues braiding described by the braid group $B_3$. Additionally, we observe chiral state transfer between three eigenstates under a fast driving scheme when no EPs are enclosed, while time-symmetric behavior occurs when at least one EP is encircled. Our findings offer insights into understanding non-Hermitian topological structures and the manipulation of quantum states through dynamic operations.

A common trait of many machine learning models is that it is often difficult to understand and explain what caused the model to produce the given output. While the explainability of neural networks has been an active field of research in the last years, comparably little is known for quantum machine learning models. Despite a few recent works analyzing some specific aspects of explainability, as of now there is no clear big picture perspective as to what can be expected from quantum learning models in terms of explainability. In this work, we address this issue by identifying promising research avenues in this direction and lining out the expected future results. We additionally propose two explanation methods designed specifically for quantum machine learning models, as first of their kind to the best of our knowledge. Next to our pre-view of the field, we compare both existing and novel methods to explain the predictions of quantum learning models. By studying explainability in quantum machine learning, we can contribute to the sustainable development of the field, preventing trust issues in the future.

Graph states are a class of important multiparty entangled states, of which bell pairs are the special case. Realizing a robust and fast distribution of arbitrary graph states in the downstream layer of the quantum network can be essential for further large-scale quantum networks. We propose a novel quantum network protocol called P2PGSD inspired by the classical Peer-to-Peer (P2P) network to efficiently implement the general graph state distribution in the network layer, which demonstrates advantages in resource efficiency and scalability over existing methods for sparse graph states. An explicit mathematical model for a general graph state distribution problem has also been constructed, above which the intractability for a wide class of resource minimization problems is proved and the optimality of the existing algorithms is discussed. In addition, we proposed the spacetime network inspired by the symmetry from relativity for memory management in network problems and used it to improve our proposed algorithm. The advantages of our protocols are confirmed by numerical simulations showing an improvement of up to 50\% for general sparse graph states, paving the way for a resource-efficient multiparty entanglement distribution across any network topology.

Binary optimization is a fundamental area in computational science, with wide-ranging applications from logistics to cryptography, where the tasks are often formulated as Quadratic or Polynomial Unconstrained Binary Optimization problems (QUBO/PUBO). In this work, we propose to use a parametrized Gaussian Boson Sampler (GBS) with threshold detectors to address such problems. We map general PUBO instance onto a quantum Hamiltonian and optimize the Conditional Value-at-Risk of its energy with respect to the GBS ansatz. In particular, we observe that, when the algorithm reduces to standard Variational Quantum Eigensolver, the cost function is analytical. Therefore, it can be computed efficiently, along with its gradient, for low-degree polynomials using only classical computing resources. Numerical experiments on 3-SAT and Graph Partitioning problems show significant performance gains over random guessing, providing a first proof of concept for our proposed approach.

Exploiting inherent symmetries is a common and effective approach to speed up the simulation of quantum systems. However, efficiently accounting for non-Abelian symmetries, such as the $SU(2)$ total-spin symmetry, remains a major challenge. In fact, expressing total-spin eigenstates in terms of the computational basis can require an exponentially large number of coefficients. In this work, we introduce a novel formalism for designing quantum algorithms directly in an eigenbasis of the total-spin operator. Our strategy relies on the symmetric group approach in conjunction with a truncation scheme for the internal degrees of freedom of total-spin eigenstates. For the case of the antiferromagnetic Heisenberg model, we show that this formalism yields a hierarchy of spin-adapted Hamiltonians, for each truncation threshold, whose ground-state energy and wave function quickly converge to their exact counterparts, calculated on the full model. These truncated Hamiltonians can be encoded with sparse and local qubit Hamiltonians that are suitable for quantum simulations. We demonstrate this by developing a state-preparation schedule to construct shallow quantum-circuit approximations, expressed in a total-spin eigenbasis, for the ground states of the Heisenberg Hamiltonian in different symmetry sectors.

Optical Schr\"{o}dinger cat states are non-Gaussian states with applications in quantum technologies, such as for building error-correcting states in quantum computing. Yet the efficient generation of high-fidelity optical Schr\"{o}dinger cat states is an outstanding problem in quantum optics. Here, we propose using squeezed superpositions of zero and two photons, $|\theta\rangle = \cos{(\theta/2)}|0\rangle + \sin{(\theta/2)}|2\rangle$, as ingredients for protocols to efficiently generate high-fidelity cat states. We present a protocol using linear optics with success probability $P\gtrsim 50\%$ that can generate cat states of size $|\alpha|^2=5$ with fidelity $F>0.99$. The protocol relies only on detecting single photons and is remarkably tolerant of loss, with $2\%$ detection loss still achieving $F>0.98$ for cats with $|\alpha|^2=5$. We also show that squeezed $\theta$ states are ideal candidates for nonlinear photon subtraction using a two-level system with near deterministic success probability and fidelity $F>0.98$ for cat states of size $|\alpha|^2=5$. Schemes for generating $\theta$ states using quantum emitters are also presented. Our protocols can be implemented with current state-of-the-art quantum optics experiments.

The main task of quantum circuit synthesis is to efficiently and accurately implement specific quantum algorithms or operations using a set of quantum gates, and optimize the circuit size. It plays a crucial role in Noisy Intermediate-Scale Quantum computation. Most prior synthesis efforts have employed CNOT or CZ gates as the 2-qubit gates. However, the SQiSW gate, also known as the square root of iSWAP gate, has garnered considerable attention due to its outstanding experimental performance with low error rates and high efficiency in 2-qubit gate synthesis. In this paper, we investigate the potential of the SQiSW gate in various synthesis problems by utilizing only the SQiSW gate along with arbitrary single-qubit gates, while optimizing the overall circuit size. For exact synthesis, the upper bound of SQiSW gates to synthesize arbitrary 3-qubit and $n$-qubit gates are 24 and $\frac{139}{192}4^n(1+o(1))$ respectively, which relies on the properties of SQiSW gate in Lie theory and quantum shannon decomposition. We also introduce an exact synthesis scheme for Toffoli gate using only 8 SQiSW gates, which is grounded in numerical observation. More generally, with respect to numerical approximations, we propose and provide a theoretical analysis of a pruning algorithm to reduce the size of the searching space in numerical experiment to $\frac{1}{12}+o(1)$ of previous size, helping us reach the result that 11 SQiSW gates are enough in arbitrary 3-qubit gates synthesis up to an acceptable numerical error.

Hardware efficient methods for high fidelity quantum state measurements are crucial for superconducting qubit experiments, as qubit numbers grow and feedback and state reset begin to be employed for quantum error correction. We present a 3D re-entrant cavity filter designed for frequency-multiplexed readout of superconducting qubits. The cavity filter is situated out of the plane of the qubit circuit and capacitively couples to an array of on-chip readout resonators in a manner that can scale to large qubit arrays. The re-entrant cavity functions as a large-linewidth bandpass filter with intrinsic Purcell filtering. We demonstrate the concept with a four-qubit multiplexed device.

Quantum computers are known for their potential to achieve up-to-exponential speedup compared to classical computers for certain problems.To exploit the advantages of quantum computers, we propose quantum algorithms for linear stochastic differential equations, utilizing the Schr\"odingerisation method for the corresponding approximate equation by treating the noise term as a (discrete-in-time) forcing term. Our algorithms are applicable to stochastic differential equations with both Gaussian noise and $\alpha$-stable L\'evy noise. The gate complexity of our algorithms exhibits an $\mathcal{O}(d\log(Nd))$ dependence on the dimensions $d$ and sample sizes $N$, where its corresponding classical counterpart requires nearly exponentially larger complexity in scenarios involving large sample sizes. In the Gaussian noise case, we show the strong convergence of first order in the mean square norm for the approximate equations. The algorithms are numerically verified for the Ornstein-Uhlenbeck processes, geometric Brownian motions, and one-dimensional L\'evy flights.

We present a scheme to realize a topological superconducting system supporting Majorana zero modes, within a number-conserving framework suitable for optical-lattice experiments. Our approach builds on the engineering of pair-hopping processes on a ladder geometry, using a sequence of pulses that activate single-particle hopping in a time-periodic manner. We demonstrate that this dynamic setting is well captured by an effective Hamiltonian that preserves the parity symmetry, a key requirement for the stabilization of Majorana zero modes. The phase diagram of our system is determined using a bosonization theory, which is then validated by a numerical study of the topological bulk gap and entanglement spectrum using matrix product states. Our results indicate that Majorana zero modes can be stabilized in a large parameter space, accessible in optical-lattice experiments.

Quantum dots (QDs) are pivotal for the development of quantum technologies, with applications ranging from single-photon sources for secure communication to quantum computing infrastructures. Understanding the electron dynamics within these QDs is essential for characterizing their properties and functionality. Here, we show how by virtue of the recently introduced quantum polyspectral analysis of transport measurements, the complex transport measurements of multi-electron QD systems can be analyzed. This method directly relates higher-order temporal correlations of a raw quantum point contact (QPC) current measurement to the Liouvillian of the measured quantum system. By applying this method to the two-level switching dynamics of a double QD system, we reveal a hidden third state, without relying on the identification of quantum jumps or prior assumptions about the number of involved quantum states. We show that the statistics of the QPC current measurement can identically be described by different three-state Markov models, each with significantly different transition rates. Furthermore, we compare our method to a traditional analysis via waiting-time distributions for which we prove that the statistics of a three-state Markov model is fully described without multi-time waiting-time distributions even in the case of two level switching dynamics. Both methods yield the same parameters with a similar accuracy. The quantum polyspectra method, however, stays applicable in scenarios with low signal-to-noise, where the traditional full counting statistics falters. Our approach challenges previous assumptions and models, offering a more nuanced understanding of QD dynamics and paving the way for the optimization of quantum devices.

We present a scheme to extend the range and precision of temperature measurements employing a qubit chain governed by Heisenberg $XX$ and Dzyaloshinskii-Moriya (DM) interactions. Our approach leverages the absence of coherences in the probe qubit's density matrix, enabling the probe to act as a detector for distinct transition frequencies within the system. By systematically tuning system parameters, we show that the number of measurable transition frequencies - and consequently, the quantum Fisher information (QFI) peaks - grows linearly with the size of the qubit chain. This linear scaling offers a scalable pathway for thermometry, allowing the measurement of a broad range of temperatures with a single probe qubit. We begin by investigating a two-qubit system coupled via the same interactions, demonstrating that the allowed energy transitions result in different temperature sensitivity profiles characterized by single and multiple peaks in QFI. Finally, we extend our analysis to a chain of an arbitrary number of ancilla qubits and find that adding more energy transitions can further widen the temperature estimation range, making it possible to estimate the ultralow temperatures through the emergence of an arbitrary number of peaks in QFI. Our findings highlight the potential of qubit chain systems as efficient and precise tools for low-temperature quantum thermometry.

We investigate the intricate dynamics of quantum coherence and non-classical correlations in a two-qubit open quantum system coupled to a squeezed thermal reservoir. By exploring the correlations between spatially separated qubits, we unravel the complex interplay between quantum correlations and decoherence induced by the reservoir. Our findings demonstrate that non-classical correlations such as quantum consonance, quantum discord, local quantum uncertainty, and quantum Fisher information are highly sensitive to the collective regime. These insights identify key parameters for optimizing quantum metrology and parameter estimation in systems exposed to environmental interactions. Furthermore, we quantify these quantum correlations in the context of practical applications such as quantum teleportation, using the two metrics viz. maximal teleportation fidelity and fidelity deviation. This work bridges theoretical advancements with real-world applications, offering a comprehensive framework for leveraging quantum resources under the influence of environmental decoherence.

Point tomography is a new approach to the problem of state estimation, which is arguably the most efficient and simple method for modern high-precision quantum information experiments. In this scenario, the experimenter knows the target state that their device should prepare, except that intrinsic systematic errors will create small discrepancies in the state actually produced. By introducing a new kind of informationally complete measurement, dubbed Fisher-symmetric measurements, point tomography determines deviations from the expected state with optimal efficiency. In this method, the number of outcomes of a measurement saturating the Gill-Massar limit for reconstructing a $d$-dimensional quantum states can be reduced from $\sim 4d-3$ to only $2d-1$ outcomes. Thus, providing better scalability as the dimension increases. Here we demonstrate the experimental viability of point tomography. Using a modern photonic platform constructed with state-of-the-art multicore optical fiber technology, we generate 4-dimensional quantum states and implement seven-outcome Fisher-symmetric measurements. Our experimental results exhibit the main feature of point tomography, namely a precision close to the Gill-Massar limit with a single few-outcome measurement. Specifically, we achieved a precision of $3.8/N$ while the Gill-Massar limit for $d=4$ is $3/N$ ($N$ being the ensemble size).

Exchange-only (EO) spin qubits in quantum dots offer an expansive design landscape for architecting scalable device layouts. The study of two-EO-qubit operations, which involve six electrons in six quantum dots, has so far been limited to a small number of the possible configurations, and previous works lack analyses of design considerations and implications for quantum error correction. Using a simple and fast optimization method, we generate complete pulse sequences for CX, CZ, iSWAP, leakage-controlled CX, and leakage-controlled CZ two-qubit gates on 450 unique planar six-dot topologies and analyze differences in sequence length (up to 43% reduction) across topology classes. In addition, we show that relaxing constraints on post-operation spin locations can yield further reductions in sequence length; conversely, constraining these locations in a particular way generates a CXSWAP operation with minimal additional cost over a standard CX. We integrate this pulse library into the Intel quantum stack and experimentally verify pulse sequences on a Tunnel Falls chip for different operations in a linear-connectivity device to confirm that they work as expected. Finally, we explore architectural implications of these results for quantum error correction. Our work guides hardware and software design choices for future implementations of scalable quantum dot architectures.

We explore the relation between a classical periodic Hamiltonian system and an associated discrete quantum system on a torus in phase space. The model is a sinusoidally perturbed Harper model and is similar to the sinusoidally perturbed pendulum. Separatrices connecting hyperbolic fixed points in the unperturbed classical system become chaotic under sinusoidal perturbation. We numerically compute eigenstates of the Floquet propagator for the associated quantum system. For each Floquet eigenstate we compute a Husimi distribution in phase space and an energy and energy dispersion from the expectation value of the unperturbed Hamiltonian operator. The Husimi distribution of each Floquet eigenstate resembles a classical orbit with a similar energy and similar energy dispersion. Chaotic orbits in the classical system are related to Floquet eigenstates that appear ergodic. For a hybrid regular and chaotic system, we use the energy dispersion to separate the Floquet eigenstates into ergodic and integrable subspaces. The distribution of quasi-energies in the ergodic subspace resembles that of a random matrix model. The width of a chaotic region in the classical system is estimated by integrating the perturbation along a separatrix orbit. We derive a related expression for the associated quantum system from the averaged perturbation in the interaction representation evaluated at states with energy close to the separatrix.

The transmission of light through an ensemble of two-level emitters in a one-dimensional geometry is commonly described by one of two emblematic models of quantum electrodynamics (QED): the driven-dissipative Dicke model or the Maxwell-Bloch equations. Both exhibit distinct features of phase transitions and phase separations, depending on system parameters such as optical depth and external drive strength. Here, we explore the crossover between these models via a parent spin model from bidirectional waveguide QED, by varying positional disorder among emitters. Solving mean-field equations and employing a second-order cumulant expansion for the unidirectional model -- equivalent to the Maxwell-Bloch equations -- we study phase diagrams, the emitter's inversion, and transmission depending on optical depth, drive strength, and spatial disorder. We find in the thermodynamic limit the emergence of phase separation with a critical value that depends on the degree of spatial order but is independent of inhomogeneous broadening effects. Even far from the thermodynamic limit, this critical value marks a special point in the emitter's correlation landscape of the unidirectional model and is also observed as a maximum in the magnitude of inelastically transmitted photons. We conclude that a large class of effective one-dimensional systems without tight control of the emitter's spatial ordering can be effectively modeled using a unidirectional waveguide approach.

Determining whether an abstract simplicial complex, a discrete object often approximating a manifold, contains multi-dimensional holes is a task deeply connected to quantum mechanics and proven to be QMA1-hard by Crichigno and Kohler. This task can be expressed in linear algebraic terms, equivalent to testing the non-triviality of the kernel of an operator known as the Combinatorial Laplacian. In this work, we explore the similarities between abstract simplicial complexes and signed or unsigned graphs, using them to map the spectral properties of the Combinatorial Laplacian to those of signed and unsigned graph Laplacians. We prove that our transformations preserve efficient sparse access to these Laplacian operators. Consequently, we show that key spectral properties, such as testing the presence of balanced components in signed graphs and the bipartite components in unsigned graphs, are QMA1-hard. These properties play a paramount role in network science. The hardness of the bipartite test is relevant in quantum Hamiltonian complexity, as another example of testing properties related to the eigenspace of a stoquastic Hamiltonians are quantumly hard in the sparse input model for the graph.

Satellite to ground quantum communication typically operates at night to reduce background signals, however it remains susceptible to noise from light pollution of the night sky. In this study we compare several methodologies for determining whether a Quantum Ground Station (QGS) site is viable for exchanging quantum signals with the upcoming Quantum Encryption and Science Satellite (QEYSSat) mission. We conducted ground site characterization studies at three locations in Canada: Waterloo, Ontario, Calgary, Alberta, and Priddis, Alberta. Using different methods we estimate the background counts expected to leak into the satellite-ground quantum channel, and determined whether the noise levels could prevent a quantum key transfer. We also investigate how satellite data recorded from the Visible Infrared Imaging Radiometer Suite (VIIRS) can help estimate conditions of a particular site, and find reasonable agreement with the locally recorded data. Our results indicate that the Waterloo, Calgary, and Priddis QGS sites should allow both quantum uplinks and downlinks with QEYSSat, despite their proximity to urban centres. Furthermore, our approach allows the use of satellite borne instrument data (VIIRS) to remotely and efficiently determine the potential of a ground site.

We propose and implement a comprehensive quantum compilation toolkit for solving the maximum independent set (MIS) problem on quantum hardware based on Rydberg atom arrays. Our end-to-end pipeline involves three core components to efficiently map generic MIS instances onto Rydberg arrays with unit-disk connectivity, with modules for graph reduction, hardware compatibility checks, and graph embedding. The first module (reducer) provides hardware-agnostic and deterministic reduction logic that iteratively reduces the problem size via lazy clique removals. We find that real-world networks can typically be reduced by orders of magnitude on sub-second time scales, thus significantly cutting down the eventual load for quantum devices. Moreover, we show that reduction techniques may be an important tool in the ongoing search for potential quantum speedups, given their ability to identify hard problem instances. In particular, for Rydberg-native MIS instances, we observe signatures of an easy-hard-easy transition and quantify a critical degree indicating the onset of a hard problem regime. The second module (compatibility checker) implements a hardware compatibility checker that quickly determines whether or not a given input graph may be compatible with the restrictions imposed by Rydberg quantum hardware. The third module (embedder) describes hardware-efficient graph embedding routines to generate (approximate) encodings with controllable overhead and optimized ancilla placements. We exemplify our pipeline with experiments run on the QuEra Aquila device available on Amazon Braket. In aggregate, our work provides a set of tools that extends the class of problems that can be tackled with near-term Rydberg atom arrays.

In quantum cryptography, fundamental laws of quantum physics are exploited to enhance the security of cryptographic tasks. Quantum key distribution is by far the most studied protocol to date, enabling the establishment of a secret key between trusted parties. However, there exist many practical use-cases in communication networks, which also involve parties in distrustful settings. The most fundamental quantum cryptographic building block in such a distrustful setting is quantum coin flipping, which provides an advantage compared to its classical equivalent. So far, few experimental studies on quantum coin flipping have been reported, all of which used probabilistic quantum light sources facing fundamental limitations. Here, we experimentally implement a quantum strong coin flipping protocol using single-photon states and demonstrate an advantage compared to both classical realizations and implementations using faint laser pulses. We achieve this by employing a state-of-the-art deterministic single-photon source based on the Purcell-enhanced emission of a semiconductor quantum dot in combination with fast polarization-state encoding enabling a quantum bit error ratio below 3%, required for the successful execution of the protocol. The reduced multi-photon emission yields a smaller bias of the coin flipping protocol compared to an attenuated laser implementation, both in simulations and in the experiment. By demonstrating a single-photon quantum advantage in a cryptographic primitive beyond QKD, our work represents a major advance towards the implementation of complex cryptographic tasks in a future quantum internet.

We propose a teleportation protocol involving beam splitting operations and binary-outcome measurements, such as parity measurements. These operations have a straightforward implementation using the dispersive regime of the Jaynes-Cummings Hamiltonian, making our protocol suitable for a broad class of platforms, including trapped ions, circuit quantum electrodynamics and acoustodynamics systems. In these platforms homodyne measurements of the bosonic modes are less natural than dispersive measurements, making standard continuous variable teleportation unsuitable. In our protocol, Alice is in possession of two bosonic modes and Bob a single mode. An entangled mode pair between Alice and Bob is created by performing a beam splitter operation on a cat state. An unknown qubit state encoded by cat states is then teleported from Alice to Bob after a beamsplitting operation, measurement sequence, and a conditional correction. In the case of multiple measurements, near-perfect fidelity can be obtained. We discuss the optimal parameters in order to maximize the fidelity under a variety of scenarios.

It is advantageous for any quantum processor to support different classes of two-qubit quantum logic gates when compiling quantum circuits, a property that is typically not seen with existing platforms. In particular, access to a gate set that includes support for the CZ-type, the iSWAP-type, and the SWAP-type families of gates, renders conversions between these gate families unnecessary during compilation as any two-qubit Clifford gate can be executed using at most one two-qubit gate from this set, plus additional single-qubit gates. We experimentally demonstrate that a SWAP gate can be decomposed into one iSWAP gate followed by one CZ gate, affirming a more efficient compilation strategy over the conventional approach that relies on three iSWAP or three CZ gates to replace a SWAP gate. Our implementation makes use of a superconducting quantum processor design based on fixed-frequency transmon qubits coupled together by a parametrically modulated tunable transmon coupler, extending this platform's native gate set so that any two-qubit Clifford unitary matrix can be realized using no more than two two-qubit gates and single-qubit gates.

Continuous-variable quantum computing utilizes continuous parameters of a quantum system to encode information, promising efficient solutions to complex problems. Trapped-ion systems provide a robust platform with long coherence times and precise qubit control, enabling the manipulation of quantum information through its motional and electronic degrees of freedom. In this work, quantum operations that can be generated in trapped-ion systems are employed to investigate applications aimed at state preparation in continuous-variable quantum systems.

Linear measurement is an important class of measurements for sensing classical signals including gravitational wave (GW), dark matter, infrared ray, rotation rate, etc. In this Letter, we focus on multiparameter linear measurement and establish a general tradeoff relation that tightly constrains the fundamental quantum limits of two independent parameters in a monochromatic classical signal detected by any linear quantum device. Such a tradeoff relation is universal and fundamental for multiparameter linear measurement since arising from Heisenberg's uncertainty principle. Compared with the Holevo Cram\'er-Rao bound, our tradeoff bound can completely identify the dependence between the attainable precision limits on estimated parameters. The dependence becomes more obvious such that the individual precision can not simultaneously reach the quantum limit as the so-called incompatible coefficient rises. Eventually, we find a necessary condition under which an optimal measurement protocol can saturate the general tradeoff relation, and show that the measurement phase can be tuned for adjusting different precision weight. This result is related to many applications, particularly detuned GW sensors for searching post-merger remnants due to the direct relation between the detuned frequency and incompatible coefficient.

Balancing high sensitivity with a broad dynamic range (DR) is a fundamental challenge in measurement science, as improving one often compromises the other. While traditional quantum metrology has prioritized enhancing local sensitivity, a large DR is crucial for applications such as atomic clocks, where extended phase interrogation times contribute to wider phase range. In this Letter, we introduce a novel quantum deamplification mechanism that extends DR at a minimal cost of sensitivity. Our approach uses two sequential spin-squeezing operations to generate and detect an entangled probe state, respectively. We demonstrate that the optimal quantum interferometer limit can be approached through two-axis counter-twisting dynamics. Further expansion of DR is possible by using sequential quantum deamplification interspersed with phase encoding processes. Additionally, we show that robustness against detection noise can be enhanced by a hybrid sensing scheme that combines quantum deamplification with quantum amplification. Our protocol is within the reach of state-of-the-art atomic-molecular-optical platforms, offering a scalable, noise-resilient pathway for entanglement-enhanced metrology.

Light-matter interaction with squeezed vacuum has received much interest for the ability to enhance the native interaction strength between an atom and a photon with a reservoir assumed to have an infinite bandwidth. Here, we study a model of parametrically driven cavity quantum electrodynamics (cavity QED) for enhancing light-matter interaction while subjected to a finite-bandwidth squeezed vacuum drive. Our method is capable of unveiling the effect of relative bandwidth as well as squeezing required to observe the anticipated anti-crossing spectrum and enhanced cooperativity without the ideal squeezed bath assumption. Furthermore, we analyze the practicality of said models when including intrinsic photon loss due to resonators imperfection. With these results, we outline the requirements for experimentally implementing an effectively squeezed bath in solid-state platforms such as InAs quantum dot cavity QED such that \textit{in situ} control and enhancement of light-matter interaction could be realized.

This thesis investigates quantum algorithms for eigenstate preparation, with a focus on solving eigenvalue problems such as the Schrodinger equation by utilizing near-term quantum computing devices. These problems are ubiquitous in several scientific fields, but more accurate solutions are specifically needed as a prerequisite for many quantum simulation tasks. To address this, we establish three methods in detail: quantum adiabatic evolution with optimal control, the Rodeo Algorithm, and the Variational Rodeo Algorithm. The first method explored is adiabatic evolution, a technique that prepares quantum states by simulating a quantum system that evolves slowly over time. The adiabatic theorem can be used to ensure that the system remains in an eigenstate throughout the process, but its implementation can often be infeasible on current quantum computing hardware. We employ a unique approach using optimal control to create custom gate operations for superconducting qubits and demonstrate the algorithm on a two-qubit IBM cloud quantum computing device. We then explore an alternative to adiabatic evolution, the Rodeo Algorithm, which offers a different approach to eigenstate preparation by using a controlled quantum evolution that selectively filters out undesired components in the wave function stored on a quantum register. We show results suggesting that this method can be effective in preparing eigenstates, but its practicality is predicated on the preparation of an initial state that has significant overlap with the desired eigenstate. To address this, we introduce the novel Variational Rodeo Algorithm, which replaces the initialization step with dynamic optimization of quantum circuit parameters to increase the success probability of the Rodeo Algorithm. The added flexibility compensates for instances in which the original algorithm can be unsuccessful, allowing for better scalability.

In the era of quantum 2.0, a key technological challenge lies in preserving coherence within quantum systems. Quantum coherence is susceptible to decoherence because of the interactions with the environment. Dephasing is a process that destroys the coherence of quantum states, leading to a loss of quantum information. In this work, we explore the dynamics of the relative entropy of coherence for tripartite pure and mixed states in the presence of structured dephasing environments at finite temperatures. Our findings demonstrate that the system's resilience to decoherence depends on the bath configuration. Specifically, when each qubit interacts with an independent environment, the dynamics differ from those observed with a shared bath. In a Markov, memoryless environment, coherence in both pure and mixed states decays, whereas coherence is preserved in the presence of reservoir memory.

Bosonic codes have seen a resurgence in interest for applications as varied as fault tolerant quantum architectures, quantum enhanced sensing, and entanglement distribution. Cat codes have been proposed as low-level elements in larger architectures, and the theory of rotationally symmetric codes more generally has been significantly expanded in the recent literature. The fault-tolerant preparation and maintenance of cat code states as a stand-alone quantum error correction scheme remains however limited by the need for robust state preparation and strong inter-mode interactions. In this work, we consider the teleportation-based correction circuit for cat code quantum error correction. We show that the class of acceptable ancillary states is broader than is typically acknowledged, and exploit this to propose the use of many-component "bridge" states which, though not themselves in the cat code space, are nonetheless capable of syndrome extraction in the regime where non-linear interactions are a limiting factor.

We design two variational algorithms to optimize specific 2-local Hamiltonians defined on graphs. Our algorithms are inspired by the Quantum Approximate Optimization Algorithm. We develop formulae to analyze the energy achieved by these algorithms with high probability over random regular graphs in the infinite-size limit, using techniques from [arXiv:2110.14206]. The complexity of evaluating these formulae scales exponentially with the number of layers of the algorithms, so our numerical evaluation is limited to a small constant number of layers. We compare these algorithms to simple classical approaches and a state-of-the-art worst-case algorithm. We find that the symmetry inherent to these specific variational algorithms presents a major \emph{obstacle} to successfully optimizing the Quantum MaxCut (QMC) Hamiltonian on general graphs. Nonetheless, the algorithms outperform known methods to optimize the EPR Hamiltonian of [arXiv:2209.02589] on random regular graphs, and the QMC Hamiltonian when the graphs are also bipartite. As a special case, we show that with just five layers of our algorithm, we can already prepare states within 1.62% error of the ground state energy for QMC on an infinite 1D ring, corresponding to the antiferromagnetic Heisenberg spin chain.

Realizing universal fault-tolerant quantum computation is a key goal in quantum information science. By encoding quantum information into logical qubits utilizing quantum error correcting codes, physical errors can be detected and corrected, enabling substantial reduction in logical error rates. However, the set of logical operations that can be easily implemented on such encoded qubits is often constrained, necessitating the use of special resource states known as 'magic states' to implement universal, classically hard circuits. A key method to prepare high-fidelity magic states is to perform 'distillation', creating them from multiple lower fidelity inputs. Here we present the experimental realization of magic state distillation with logical qubits on a neutral-atom quantum computer. Our approach makes use of a dynamically reconfigurable architecture to encode and perform quantum operations on many logical qubits in parallel. We demonstrate the distillation of magic states encoded in d=3 and d=5 color codes, observing improvements of the logical fidelity of the output magic states compared to the input logical magic states. These experiments demonstrate a key building block of universal fault-tolerant quantum computation, and represent an important step towards large-scale logical quantum processors.

We focus on the optimization of neutral atom transport and transfer between optical tweezers, both critical steps towards the implementation of quantum processors and simulators. We consider four different types of experimentally relevant pulses: piece-wise linear, piece-wise quadratic, minimum jerk, and a family of hybrid linear and minimum jerk ramps. We also develop a protocol using Shortcuts to Adiabaticity (STA) techniques that allows us to include the effects of static traps. By computing a measure of the error after transport and two measures of the heating for transient times, we provide a systematic characterization of the performance of all the considered pulses and show that our proposed STA protocol outperforms the experimentally inspired pulses. After pulse shape optimization we find a lower threshold for the total time of the protocol that is compatible with the limit below which the increase in the vibrational excitations exceeds half of the amount of states hosted by the moving tweezer. Since the obtained lower bound for the atom capturing or releasing stage is 9 times faster than the one reported in state-of-the-art experiments, we interpret our results as a wake-up call towards the importance of the inclusion and optimization of the transfer between tweezers, which may be the largest bottleneck to speed. For the two pulses having the best performance (minimum jerk and STA), we determine optimal regions in the experimentally accessible parameters to implement high fidelity transport pulses. Finally, our STA results prove that a modulation in the depth of the moving tweezer designed to counteract the effect of the static traps reduces errors and allows for shorter pulse duration. To motivate the use of our STA pulse in future experiments, we provide a simple analytical approximation for the tweezer position and depth controls.

Understanding the Page curve and resolving the black hole information puzzle in terms of the entanglement dynamics of black holes has been a key question in fundamental physics. In principle, the current quantum computing can provide insights into the entanglement dynamics of black holes within some simplified models. In this regard, we utilize quantum computers to investigate the entropy of Hawking radiation using the qubit transport model, a toy qubit model of black hole evaporation. Specifically, we implement the quantum simulation of the scrambling dynamics in black holes using an efficient random unitary circuit. Furthermore, we employ the swap-based many-body interference protocol for the first time and the randomized measurement protocol to measure the entanglement entropy of Hawking radiation qubits in IBM's superconducting quantum computers. Our findings indicate that while both entanglement entropy measurement protocols accurately estimate the R\'enyi entropy in numerical simulation, the randomized measurement protocol has a particular advantage over the swap-based many-body interference protocol in IBM's superconducting quantum computers. Finally, by incorporating quantum error mitigation techniques, we establish that the current quantum computers are robust tools for measuring the entanglement entropy of complex quantum systems and can probe black hole dynamics within simplified toy qubit models.

We show that universal quantum computation can be made fault-tolerant in a scenario where the error-correction is implemented without mid-circuit measurements. To this end, we introduce a measurement-free deformation protocol of the Bacon-Shor code to realize a logical $\mathit{CCZ}$ gate, enabling a universal set of fault-tolerant operations. Independently, we demonstrate that certain stabilizer codes can be concatenated in a measurement-free way without having to rely on a universal logical gate set. This is achieved by means of the disposable Toffoli gadget, which realizes the feedback operation in a resource-efficient way. For the purpose of benchmarking the proposed protocols with circuit-level noise, we implement an efficient method to simulate non-Clifford circuits consisting of few Hadamard gates. In particular, our findings support that below-breakeven logical performance is achievable with a circuit-level error rate below $10^{-3}$. Altogether, the deformation protocol and the Toffoli gadget provide a blueprint for a fully fault-tolerant architecture without any feed-forward operation, which is particularly suited for state-of-the-art neutral-atom platforms.

Quantum Frequency Conversion (QFC) is a widely used technique to interface atomic systems with the telecom band in order to facilitate propagation over longer distances in fiber. Here we demonstrate the difference-frequency conversion from 606 nm to 1552 nm of microsecond-long weak coherent pulses at the single photon level compatible with Pr$^{3+}$:Y$_2$SiO$_5\,$ quantum memories, with high-signal to noise ratio. We use a single step difference frequency generation process with a continuous-wave pump at 994 nm in a MgO:ppLN-waveguide and ultra-narrow spectral filtering down to a bandwidth of 12.5 MHz. With this setup, we achieve the conversion of weak coherent pulses of duration up to 13.6 $\mu s$ with a device efficiency of about 25% and a signal-to-noise ratio >460 for 10 $\mu s$-long pulses containing one photon on average. This signal-to-noise ratio is large enough to enable a high-fidelity conversion of qubits emitted from an emissive quantum memory based on Pr$^{3+}$:Y$_2$SiO$_5\,$ and to realize an interface with quantum processing nodes based on narrow-linewidth cavity-enhanced trapped ions.

With the development of any quantum technology comes a need for precise control of quantum systems. Here, we evaluate the impact of control noise on a quantum Otto cycle. Whilst it is postulated that noiseless quantum engines can approach maximal Otto efficiency in finite times, the existence of white noise on the controls is shown to negatively affect average engine performance. Two methods of quantum enhancement, counterdiabatic driving and quantum lubrication, are implemented and found to improve the performance of the noisy cycle only in specified parameter regimes. To gain insight into performance fluctuations, projective energy measurements are used to construct a noise-averaged probability distribution without assuming full thermalisation or adiabaticity. From this, the variances in thermodynamic currents are observed to increase as average power and efficiency improve, and are also shown to be consistent with known bounds from thermodynamic uncertainty relations. Lastly, by comparing the average functioning of the unmonitored engine to a projectively-measured engine cycle, the role of coherence in work extraction for this quantum engine model is investigated.

Topological zero modes in topological insulators or superconductors are exponentially localized at the phase transition between a topologically trivial and nontrivial phase. These modes are solutions of a Jackiw-Rebbi equation modified with an additional term which is quadratic in the momentum. Moreover, localized fermionic modes can also be induced by harmonic potentials in superfluids and superconductors or in atomic nuclei. Here, by using inverse methods, we consider in the same framework exponentially-localized zero modes, as well as gaussian modes induced by harmonic potentials (with superexponential decay) and polynomially decaying modes (with subexponential decay), and derive the explicit and analytical form of the modified Jackiw-Rebbi equation (and of the Schr\"odinger equation) which admits these modes as solutions. We find that the asymptotic behavior of the mass term is crucial in determining the decay properties of the modes. Furthermore, these considerations naturally extend to the nonhermitian regime. These findings allow us to classify and understand topological and nontopological boundary modes in topological insulators and superconductors.

Deeply inelastic scattering (DIS) is an essential process for exploring the structure of visible matter and testing the standard model. At the same time, the theoretical interpretation of DIS measurements depends on QCD factorization theorems whose validity deteriorates at the lower values of $Q^2$ and $W^2$ typical of neutrino DIS in accelerator-based oscillation searches. For this reason, progress in understanding the origin and limits of QCD factorization is invaluable to the accuracy and precision of predictions for these upcoming neutrino experiments. In these short proceedings, we introduce a novel approach based on the quantum entropy associated with continuous distributions in QCD, using it to characterize the limits of factorization theorems relevant for the description of neutrino DIS. This work suggests an additional avenue for dissecting factorization-breaking dynamics through the quantum entropy, which could also play a role in quantum simulations of related systems.

The Sachdev-Ye-Kitaev (SYK) model is a cornerstone in the study of quantum chaos and holographic quantum matter. Real-world implementations, however, deviate from the idealized all-to-all connectivity, raising questions about the robustness of its chaotic properties. In this work, we investigate a quadratic SYK model with distance-dependent interactions governed by a power-law decay. By analytically and numerically studying the spectral form factor (SFF), we uncover how the single particle transitions manifest in the many-body system. While the SFF demonstrates robustness under slightly reduced interaction ranges, further suppression leads to a breakdown of perturbation theory and new spectral regimes, marked by a higher dip and the emergence of a secondary plateau. Our results highlight the interplay between single-particle criticality and many-body dynamics, offering new insights into the quantum chaos-to-localization transition and its reflection in spectral statistics.

Atom interferometers offer exceptional sensitivity to ultra-light dark matter (ULDM) through their precise measurement of phenomena acting on atoms. While previous work has established their capability to detect scalar and vector ULDM, their potential for detecting spin-2 ULDM remains unexplored. This work investigates the sensitivity of atom interferometers to spin-2 ULDM by considering several frameworks for massive gravity: a Lorentz-invariant Fierz-Pauli case and two Lorentz-violating scenarios. We find that coherent oscillations of the spin-2 ULDM field induce a measurable phase shift through three distinct channels: coupling of the scalar mode to atomic energy levels, and vector and tensor effects that modify the propagation of atoms and light. Atom interferometers uniquely probe all of these effects, while providing sensitivity to a different mass range from laser interferometers. Our results demonstrate the potential of atom interferometers to advance the search for spin-2 dark matter through accessing unexplored parameter space and uncovering new interactions between ULDM and atoms.

There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. However, the theory of fracton orders remains in its early stages, and the complex landscape of examples is far from being mapped out. Here we introduce the class of planon-modular (p-modular) fracton orders, a relatively simple yet still rich class of quantum orders that encompasses several well-known examples of type I fracton order. The defining property is that any non-trivial point-like excitation can be detected by braiding with planons. From this definition, we uncover a significant amount of general structure, including the assignment of a natural number (dubbed the weight) to each excitation of a p-modular fracton order. We identify simple new phase invariants, some of which are based on weight, which can easily be used to compare and distinguish different fracton orders. We also study entanglement renormalization group (RG) flows of p-modular fracton orders, establishing a close connection with foliated RG. We illustrate our general results with an analysis of several exactly solvable fracton models that we show to realize p-modular fracton orders, including Z_n versions of the X-cube, anisotropic, checkerboard, 4-planar X-cube and four color cube (FCC) models. We show that each of these models is p-modular and compute its phase invariants. We also show that each example admits a foliated RG at the level of its non-trivial excitations, which is a new result for the 4-planar X-cube and FCC models. We show that the Z_2 FCC model is not a stack of other better-studied models, but predict that the Z_n FCC model with n odd is a stack of 10 4-planar X-cubes, possibly plus decoupled layers of 2d toric code. We also show that the Z_n checkerboard model for n odd is a stack of three anisotropic models.

The density-density response in optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ has recently been shown to exhibit conformal symmetry. Using, the experimentally inferred conformal dynamic susceptibility, we compute the resultant quantum Fisher information (QFI), a witness to multi-partite entanglement. In contrast to a Fermi liquid in which the QFI is approximately temperature independent much below the Fermi energy scale, we find that the QFI increases as a power law at low temperatures but ultimately extrapolates to a constant at $T=0$. The constant is of the form, $\omega_g^{2\Delta}$, where $\Delta$ is the conformal dimension and $\omega_g$ is the UV cutoff which is on the order of the pseudogap. As this constant {depends on both UV and IR properties}, it illustrates that multipartite entanglement in a strange metal exhibits UV-IR mixing, a benchmark feature of doped Mott insulators as exemplified by dynamical spectral weight transfer. We conclude with a discussion of the implication of our results for low-energy reductions of the Hubbard model.

Coherent control of atomic and molecular scattering relies on the preparation of colliding particles in superpositions of internal states, establishing interfering pathways that can be used to tune the outcome of a scattering process. However, incoherent addition of different partial wave contributions to the integral cross sections (partial wave scrambling), commonly encountered in systems with complex collisional dynamics, poses a significant challenge, often limiting control. This work demonstrates that time-reversal symmetry can overcome these limitations by constraining the relative phases of S-matrix elements, thereby protecting coherent control against partial wave scrambling, even for collisions mediated by highly anisotropic interactions. Using the example of ultracold O$_2$-O$_2$ scattering, we show that coherent control is robust against short-range dynamical complexity. Furthermore, the time-reversal symmetry also protects the control against a distribution of collisional energy. These findings show that ultracold scattering into the final states that are time-reversal-invariant, such as the J = 0, M = 0 rotational state, can always be optimally controlled by using time-reversal-invariant initial superpositions. Beyond the ultracold regime, we observe significant differences in the controllability of crossed-molecular beam vs. trap experiments with the former being easier to control, emphasizing the cooperative role of time-reversal and permutation symmetries in maintaining control at any temperature. These results open new avenues for the coherent control of complex inelastic collisions and chemical reactions both in and outside of the ultracold regime.

In this research, we investigate second-order sideband generation (SSG) and slow-fast light using a hybrid system comprised of two coupled opto- and magnomechanical microspheres, namely a YIG sphere and a silica sphere. The YIG sphere hosts a magnon mode and a vibration mode induced by magnetostriction, whereas the silica sphere has an optical whispering gallery mode and a mechanical mode coupled via optomechanical interaction. The mechanical modes of both spheres are close in frequency and are coherently coupled by the straightway physical contact between the two microspheres. We use a perturbation approach to solve the Heisenberg-Langevin equations, offering an analytical framework for transmission rate and SSG. Using experimentally feasible settings, we demonstrate that the transmission rate and SSG are strongly dependent on the magnomechanical, optomechanical, and mechanics mechanics coupling strengths (MMCS) between the two microspheres. The numerical results show that increasing the MMCS can enhance both the transmission rate and SSG efficiency, resulting in gain within our system. Our findings, in particular, reveal that the efficiency of the SSG can be effectively controlled by cavity detuning, decay rate, and pump power. Notably, our findings suggest that modifying the system parameters can alter the group delay, thereby regulating the transition between fast and slow light propagation, and vice versa. Our protocol provides guidelines for manipulating nonlinear optical properties and controlling light propagation, with applications including optical switching, information storage, and precise measurement of weak signals.

Boundaries in gauge theory and gravity give rise to symmetries and charges at both finite and asymptotic distance. Due to their structural similarities, it is often held that soft modes are some kind of asymptotic limit of edge modes. Here, we show in Maxwell theory that there is an arguably more interesting relationship between the \emph{asymptotic} symmetries and their charges, on one hand, and their \emph{finite-distance} counterparts, on the other, without the need of a limit. Key to this observation is to embed the finite region in the global spacetime and identify edge modes as dynamical $\rm{U}(1)$-reference frames for dressing subregion variables. Distinguishing \emph{intrinsic} and \emph{extrinsic} frames, according to whether they are built from field content in- or outside the region, we find that non-trivial corner symmetries arise only for extrinsic frames. Further, the asymptotic-to-finite relation requires asymptotically charged ones (like Wilson lines). Such frames, called \emph{soft edges}, extend to asymptotia and realize the corner charge algebra by ``pulling in'' the asymptotic one from infinity. Realizing an infinite-dimensional algebra requires a new set of \emph{soft boundary conditions}, relying on the distinction between extrinsic and intrinsic data. We identify the subregion Goldstone mode as the relational observable between extrinsic and intrinsic frames and clarify the meaning of vacuum degeneracy. We also connect the asymptotic memory effect with a more operational \emph{quasi-local} one. A main conclusion is that the relationship between asymptotia and finite distance is \emph{frame-dependent}; each choice of soft edge mode probes distinct cross-boundary data of the global theory. Our work combines the study of boundary symmetries with the program of dynamical reference frames and we anticipate that core insights extend to Yang-Mills theory and gravity.

Non-reciprocal hopping induces a synthetic magnetic flux which leads to the non-Hermitian Aharonov-Bohm effect. Since non-Hermitian Hamiltonians possess both real and imaginary eigenvalues, this effect allows the observation of real and imaginary persistent currents in a ring threaded by the synthetic flux~\cite{nrh8}. Motivated by this, we investigate the behavior of persistent currents in a disordered Hatano-Nelson ring with anti-Hermitian intradimer hopping. The disorder is diagonal and we explore three distinct models, namely the Aubry-Andr\'{e}-Harper model, the Fibonacci model, both representing correlated disorder, and an uncorrelated (random) model. We conduct a detailed analysis of the energy spectrum and examine the real and imaginary parts of the persistent current under various conditions such as different ring sizes and filling factors. Interestingly, we find that real and imaginary persistent currents exhibit amplification in the presence of correlated disorder. This amplification is also observed in certain individual random configurations but vanishes after configuration averaging. Additionally, we observe both diamagnetic and paramagnetic responses in the current behavior and investigate aspects of persistent currents in the absence of disorder that have not been previously explored. Interestingly, we find that the intradimer bonds host only imaginary currents, while the interdimer bonds carry only real currents.

We construct relational observables in group field theory (GFT) in terms of covariant positive operator-valued measures (POVMs), using techniques developed in the context of quantum reference frames. We focus on matter quantum reference frames; this can be generalized to other types of frames within the same POVM-based framework. The resulting family of relational observables provides a covariant framework to extract localized observables from GFT, which is typically defined in a perspective-neutral way. Then, we compare this formalism with previous proposals for relational observables in GFT. We find that our quantum reference frame-based relational observables overcome the intrinsic limitations of previous proposals while reproducing the same continuum limit results concerning expectation values of the number and volume operators on coherent states. Nonetheless, there can be important differences for more complex operators, as well as for other types of GFT states. Finally, we also use a specific class of POVMs to show how to project states and operators from the more general perspective-neutral GFT Fock space to a perspective-dependent one where a scalar matter field plays the role of a relational clock.

The energy level statistics of uniform random graphs are studied, by treating the graphs as random tight-binding lattices. The inherent random geometry of the graphs and their dynamical spatial dimensionality, leads to various quantum chaotic and localized phases and transitions between them. Essentially the random geometry acts as disorder, whose strength is characterized by the ratio of edges over vertices R in the graphs. For dense graphs, with large ratio R, the spacing between successive energy levels follows the Wigner-Dyson distribution, leading to a quantum chaotic behavior and a metallic phase, characterized by level repulsion. For ratios near R=0.5, where a large dominating component in the graph appears, the level spacing follows the Poisson distribution with level crossings and a localized phase for the respective wavefunctions lying on the graph. For intermediate ratios R we observe a phase transition between the quantum chaotic and localized phases characterized by a semi-Poisson distribution. The values R of the critical regime where the phase transition occurs depend on the energy of the system. Our analysis shows that physical systems with random geometry, for example ones with a fluctuating/dynamical spatial dimension, contain novel universal phase transition properties, similar to those occuring in more traditional phase transitions based on symmetry breaking mechanisms, whose universal properties are strongly determined by the dimensionality of the system.

Dispersive readout of superconducting qubits is often limited by readout-drive-induced transitions between qubit levels. While there is a growing understanding of such effects in transmon qubits, the case of highly nonlinear fluxonium qubits is more complex. We theoretically analyze measurement-induced state transitions (MIST) during the dispersive readout of a fluxonium qubit. We focus on a new mechanism: a simultaneous transition/excitation involving the qubit and an internal mode of the Josephson junction array in the fluxonium circuit. Using an adiabatic Floquet approach, we show that these new kinds of MIST processes can be relevant when using realistic circuit parameters and relatively low readout drive powers. They also contribute to excess qubit dephasing even after a measurement is complete. In addition to outlining basic mechanisms, we also investigate the dependence of such transitions on the circuit parameters. We find that with a judicious choice of frequency allocations or coupling strengths, these parasitic processes can most likely be avoided.

Neutral resonant states of molecules play a very important role in the dissociation dynamics and other electronic processes that occur via intermediate capture into these states. With the goal of identifying resonant states, and their corresponding widths, of the imidogen molecule NH as a function of internuclear distance, we have performed detailed R-matrix calculations on the e + NH+ system. In a previous work, we had identified bound states of NH and Feshbach resonances in the e + NH+ system at a single geometry, namely the NH+ equilibrium Re = 2.0205 a0 . Here we present a much more detailed work by repeating the calculation on over 60 internuclear distances to obtain the corresponding potential energy curves. The bound states for nine symmetries have been detailed many of which, particularly the singlet states, were never studied before. Several resonant states of different symmetries, which were unknown until now, have been systematically identified and their widths calculated in the present work, which proved much more challenging due to presence of many avoided crossings. It is hoped that the bound and the new resonant states obtained here will open up other molecular dynamics studies, since for several dissociative processes, although experimental data existed for more than a decade, these are still uncorroborated due to absence of molecular data, and hence subsequent theoretical calculations.

Two-dimensional optical spectroscopy experiments have shown that exciton transfer pathways in the Fenna-Matthews-Olson (FMO) photosynthetic complex differ drastically under reduced and oxidised conditions, suggesting a functional role for collective vibronic mechanisms that may be active in the reduced form but attenuated in the oxidised state. Higgins et al. [PNAS 118 (11) e2018240118 (2021)] used Redfield theory to link the experimental observations to altered exciton transfer rates due to oxidative onsite energy shifts that detune excitonic energy gaps from a specific vibrational frequency of the bacteriochlorophyll (BChl) a. Using a memory kernel formulation of the hierarchical equations of motion, we present non-perturbative estimations of transfer rates that yield a modified physical picture. Our findings indicate that onsite energy shifts alone cannot reproduce the observed rate changes in oxidative environments, either qualitatively or quantitatively. By systematically examining combined changes both in site energies and the local environment for the oxidised complex, while maintaining consistency with absorption spectra, our results suggest that vibronic tuning of transfer rates may indeed be active in the reduced complex. However, we achieve qualitative, but not quantitative, agreement with the experimentally measured rates. Our analysis indicates potential limitations of the FMO electronic Hamiltonian, which was originally derived by fitting spectra to second-order cumulant and Redfield theories. This suggests that reassessment of these electronic parameters with a non-perturbative scheme, or derived from first principles, is essential for a consistent and accurate understanding of exciton dynamics in FMO under varying redox conditions.

We present a new approach to investigating Rydberg molecules by demonstrating the formation and characterization of individual Rb$^{*}$Cs Rydberg molecules using optical tweezers. By employing single-atom detection of Rb and Cs, we observe molecule formation via correlated loss of both species and study the formation dynamics with single-particle resolution. We control the interatomic distances by manipulating the relative wavefunction of atom pairs using the tweezer intensity, optimizing the coupling to molecular states and exploring the effect of the tweezer on these states. Additionally, we demonstrate molecule association with atoms trapped in separate tweezers, paving the way for state-selective assembly of polyatomic molecules. The observed binding energies, molecular alignment, and bond lengths are in good agreement with theory. Our approach is broadly applicable to Rydberg tweezer platforms, expanding the range of available molecular systems and enabling the integration of Rydberg molecules into existing quantum science platforms.

We propose a novel and systematic recurrence method for the energy spectra of non-Hermitian systems under open boundary conditions based on the recurrence relations of their characteristic polynomials. Our formalism exhibits better accuracy and performance on multi-band non-Hermitian systems than numerical diagonalization or the non-Bloch band theory. It also provides a targeted and efficient formulation for the non-Hermitian edge spectra. As demonstrations, we derive general expressions for both the bulk and edge spectra of multi-band non-Hermitian models with nearest-neighbor hopping and under open boundary conditions, such as the non-Hermitian Su-Schrieffer-Heeger and Rice-Mele models and the non-Hermitian Hofstadter butterfly - 2D lattice models in the presence of non-reciprocity and perpendicular magnetic fields, which is only made possible by the significantly lower complexity of the recurrence method. In addition, we use the recurrence method to study non-Hermitian edge physics, including the size-parity effect and the stability of the topological edge modes against boundary perturbations. Our recurrence method offers a novel and favorable formalism to the intriguing physics of non-Hermitian systems under open boundary conditions.

Photocurrent-induced harmonics appear in gases and solids due to tunnel ionization of electrons in strong fields and subsequent acceleration. In contrast to three-step harmonic emission, no return to the parent ions is necessary. Here we show that the same mechanism produces harmonics in metallic nanostructures in strong fields. Furthermore, we demonstrate how strong local field gradient, appearing as a consequence of the field enhancement, affects photocurrent-induced harmonics. This influence can shed light at the state of electron as it appears in the continuum, in particular, to its initial velocity.

Categorical symmetries have recently been shown to generalize the classification of phases of matter, significantly broadening the traditional Landau paradigm. To test these predictions, we propose a simple spin chain model that encompasses all gapped phases and second-order phase transitions governed by the categorical symmetry $\mathsf{Rep}(D_8)$. This model not only captures the essential features of non-invertible phases but is also straightforward enough to enable practical realization. Specifically, we outline an implementation using neutral atoms trapped in optical tweezer arrays. Employing a dual-species setup and Rydberg blockade, we propose a digital simulation approach that can efficiently implement the many-body evolution in several nontrivial quantum phases.

The quantum extremal island rule allows us to compute the Page curves of Hawking radiation in semi-classical gravity. In this work, we study the connection between these calculations and the thermalisation of chaotic quantum many-body systems, using a coarse-grained description of entanglement dynamics known as the entanglement membrane. Starting from a double-holographic model of eternal two-sided asymptotically AdS$_d$ ($d>2$) black hole each coupled to a flat $d$-dimensional bath, we show that the entanglement dynamics in the late-time, large-subregion limit is described by entanglement membrane, thereby establishing a quantitative equivalence between a semi-classical gravity and a chaotic quantum many-body system calculation of the Page curve.

The rotational states of ultracold polar molecules possess long radiative lifetimes, microwave-domain coupling, and tunable dipolar interactions. Coherent dynamics between pairs of rotational states have been used to demonstrate simple models of quantum magnetism and to manipulate quantum information stored as qubits. The availability of numerous rotational states has led to many proposals to implement more complicated models of quantum magnetism, higher-dimensional qudits, and intricate state networks as synthetic dimensions; however, these are yet to be experimentally realised. The primary issue limiting their implementation is the detrimental effect of the optical trapping environment on coherence, which is not easily mitigated for systems beyond two levels. To address this challenge, we investigate the applicability of magic-wavelength optical tweezer traps to facilitate multitransition coherence between rotational states. We demonstrate simultaneous second-scale coherence between three rotational states. Utilising this extended coherence, we perform multiparameter estimation using a generalised Ramsey sequence and demonstrate coherent spin-1 dynamics encoded in the rotational states. Our work paves the way to implementing proposed quantum simulation, computation, and metrology schemes that exploit the rich rotational structure of ultracold polar molecules.

It has been conjectured that the size of the black hole interior captures the quantum gate complexity of the underlying boundary evolution. In this short note we aim to provide a further microscopic evidence for this by directly relating the area of a certain codimension-two surface traversing the interior to the depth of the quantum circuit. Our arguments are based on establishing such relation rigorously at early times using the notion of operator Schmidt rank and then extrapolating it to later times by mapping bulk surfaces to cuts in the circuit representation.