1 Introduction↩︎

Encoding classical data into quantum systems is the first step in virtually every quantum computing and quantum communication protocol. In quantum machine learning [1], raw data must be mapped to quantum states before being fed into parameterized quantum circuits. In quantum key distribution [2], bits are encoded in non-orthogonal quantum states to guarantee information-theoretic security. In quantum communication [3], code-words must be chosen to maximize fidelity at the receiver. In all these settings the choice of encoding is consequential. Not all encodings are equal.

A systematic theory of optimal quantum encoding has remained elusive, partly because “optimal” depends heavily on the figure of merit. Prior work has studied optimal encoding for fidelity [3], for retrieval [4], for communication over specific channel families [5], for security under gentle measurements [6], and for incompatibility-based key sharing [7]. However, a universal, task-independent figure of merit for quantum encoders, i.e., an encoding that simultaneously bounds performance across a wide class of inference problems, has not previously been identified.

In this paper, we adopt maximal quantum leakage [8] as that figure of merit, and pursue it to a complete characterization of the optimal encoder. Maximal quantum leakage was introduced in [8] as the largest multiplicative increase in an adversary’s guessing probability that can result from any measurement on the quantum encoding of a classical random variable. This notion relates to measured Sibson mutual information of order infinity. It satisfies all the axiomatic requirements of a rigorous information-leakage measure, i.e., positivity, independence, and the post-processing inequality, and is independent of both the distribution of \(X\) and the specific inference task.

To justify the choice of maximal quantum leakage as a figure of merit for quantum encoding, we prove that the accuracy of any statistical inference problem of any quantum inference procedure is bounded above by the maximal quantum leakage. This bound is in fact tight in the sense that there exists at least one inference problem that saturates the presented upper bound. Maximal quantum leakage depends only on the encoding and not on the inference task, which renders its maximizer the universal optimal encoder for quantum inference algorithms. We subsequently prove that maximal quantum leakage is equal to the optimal success probability in minimum-error quantum state discrimination with equal priors. Maximizing leakage is therefore equivalent to designing codewords that are maximally distinguishable. We prove that pure states are optimal for encoding under the developed figure merit and characterize the optimal encoding for a range of parameters. An important observation is that tight frames are optimal when the dimension of the quantum system is small enough. Among all tight frames equiangular tight frames (ETFs) have the smallest overlap with each other and saturate the Welch bound on all pairwise overlaps. ETFs are the most symmetric and most robust optimal encodings. We provide closed-form expressions for a range of parameters.

The connection between tight frames and quantum state discrimination has appeared in the frame-theory literature [9], [10], but its application to the design of quantum codewords for a universal inference criterion is new to this paper.

The rest of the paper is organized as follows. Section 2 establishes notation and definitions. Section 3 presents the statistical inference model, the universal accuracy bound, and the relationship with state discrimination. Section 4 presents the optimal encoding. Section 5 establishes the relationship with tight frames and presents explicit constructions for a range of parameters. Section 6 reports numerical examples. Section 7 concludes the paper.

2 Preliminaries↩︎

2.0.0.1 Notation.

All logarithms are in binary basis, motivated by the notion of bits in information theory. Random variables are denoted by capital Roman letters, e.g., \(X\). A discrete random variable \(X\) with finite alphabet \(\mathcal{X}\) is characterized by its probability mass function \(\mathbb{P}\{X=x\}>0\) for \(x\in\mathcal{X}\). The restriction to \(\mathbb{P}\{X=x\}>0\), \(\forall x\in\mathcal{X}\), is without loss of generality as any realization with zero probability can be removed with no impact. We write \(|\mathcal{X}|\) to denote the number of distinct alphabets or the cardinality of the set \(\mathcal{X}\).

2.0.0.2 Quantum states and measurements.

Let \(\mathcal{H}\) denote a finite-dimensional complex Hilbert space of dimension \(d:=\dim\mathcal{H}\). The set of linear operators on \(\mathcal{H}\) is \(\mathcal{L}(\mathcal{H})\). A density operator is a positive semi-definite operator \(\rho\in\mathcal{L}(\mathcal{H})\) with \(\trace(\rho)=1\); the set of all density operators is \(\mathcal{S}(\mathcal{H})\). A density operator \(\rho\) is pure if \(\rank(\rho)=1\), equivalently if \(\rho= \ket{\psi} \bra{\psi}\) for some unit vector \(\ket{\psi}\in\mathcal{H}\). A positive operator-valued measure (POVM) is a finite collection \(\{F_y \}_{y\in\mathcal{Y}} \subset\mathcal{L}(\mathcal{H})\) satisfying \(F_y\geq 0\), \(\forall y\in\mathcal{Y}\), and \(\sum_{y\in\mathcal{Y}}F_y=I\). By Born’s rule, the probability of outcome \(y\) when measuring state \(\rho\) is \(\mathbb{P}\{Y=y\} =\trace(F_y\rho)\). A quantum channel \(\mathcal{N}:\mathcal{S}(\mathcal{H})\to\mathcal{S}(\mathcal{H}')\) is a completely positive and trace-preserving linear map [11].

2.0.0.3 Maximal quantum leakage.

Consider encoding classical data \(X\in\mathcal{X}\) into a quantum system \(A\) by preparing the system in state \(\rho^x\in \mathcal{S}(\mathcal{H})\) when \(X=x\), \(\forall x\in\mathcal{X}\). The collection \(\mathcal{R}=\{\rho^x\}_{x\in\mathcal{X}}\) is called the quantum encoding of \(X\).

Maximal quantum leakage captures the largest multiplicative increase in the probability of correctly guessing an arbitrary function of \(X\) that any measurement on the quantum encoding can produce [8]. It satisfies the post-processing inequality, i.e., \(\mathcal{Q}(X\to A)_{\mathcal{N}(\rho)}\leq \mathcal{Q}(X \to A)_\rho\) for any quantum channel \(\mathcal{N}\) [8]. Furthermore, maximal quantum leakage is known to be upper bounded as \(\mathcal{Q}(X \to A)_\rho\leq\min\{\log (N),2\log (d)\}\) [8].

2.0.0.4 Minimum-error quantum state discrimination.

Consider a quantum system prepared in state \(\rho_i\) with prior probability \(q_i\), \(i=1,\dots,N\). A POVM \(\{M_i\}_{i=1}^N\) is applied to identify the state with the probability of success given by \(\sum_i q_i\trace(M_i\rho_i)\). The optimal success probability is \[\begin{align} \label{eqn:pguess} P_{\rm guess}\left( \{q_i,\rho_i\}_{i=1}^N\right) \;:=\; \max_{\{M_i\}}\sum_{i=1}^N q_i\trace(M_i\rho_i) \quad\text{s.t.}\quad M_i\geq 0,\;\;\sum_{i=1}^N M_i=I. \end{align}\tag{1}\] This semi-definite program is efficiently solvable and admits strong duality [12], [13].

2.0.0.5 Frames.

A collection of vectors \(\{\ket{\psi_x}\}_{x\in\mathcal{X}}\) in \(\mathcal{H}\) is a frame if it spans \(\mathcal{H}\), i.e., if the frame operator \(S := \sum_{x\in\mathcal{X}} \ket{\psi_x}\bra{\psi_x}\) is invertible. The frame is tight with frame bound \(A\) if \(S=AI\), and unit-norm tight if additionally each \(\ket{\psi_x}\) is a unit vector and \(A = N/d\). Throughout this paper, “tight frame” means unit-norm tight frame unless stated otherwise.

3 Quantum-Assisted Statistical Inference↩︎

Consider jointly distributed discrete random variables \(X\in\mathcal{X}\) (input) and \(Z\in\mathcal{Z}\) (output), with the inference goal of predicting \(Z\) from \(X\). This is a general framework capturing, as an example, quantum machine learning (for classification) and quantum sensing (in discretized form). The information pipeline is depicted in Figure 1 with its building elements discussed below. A quantum inference procedure \((\mathcal{R},\mathcal{N},\mathcal{F},\gamma)\) comprises:

• Encoding: For each realization \(X=x\), prepare system \(A\) in state \(\rho^x\in\mathcal{S}(\mathcal{H})\) with \(\dim(\mathcal{H})=d\), giving encoding \(\mathcal{R}=\{\rho^x\}_{x\in\mathcal{X}}\);

• Processing: Apply quantum channel \(\mathcal{N}:\mathcal{S}(\mathcal{H})\to\mathcal{S}(\mathcal{H}')\), which can be a quantum machine learning policy with parameters to be optimized or a specific algorithm, such as quantum Fourier transform;

• Measurement: Apply POVM \(\mathcal{F}=\{F_y\}_{y\in\mathcal{Y}}\) on system \(A\), yielding outcome \(Y\in\mathcal{Y}\) with probability \(\mathbb{P}\{Y=y|X=x\}=\trace(F_y\mathcal{N}(\rho^x))\);

• Classical post-processing: Form estimate \(\widehat{Z}\in\mathcal{Z}\) from \(Y\) via any stochastic kernel \(\gamma_{zy}=\mathbb{P}\{\widehat{Z}=z|Y=y\}\), such as classical machine learning or signal processing algorithms.

The accuracy of the procedure is \(\mathbb{P}\{\widehat{Z}=Z\}\). This is the probability of correct inference of output \(Z\) based on input \(X\). In what follows, we use the notation \(N:=|\mathcal{X}|\) and \(d:=\dim(\mathcal{H})\) for notational brevity when needed.

3.0.0.1 Equivalence with minimum-error discrimination.

The following theorem recasts the performance bound in Theorem 1 in terms of quantum state discrimination. A version of this result, for general prior distributions, is established in [15]. Here, we give an independent self-contained proof for the uniform-prior case relevant here.

Note that, in Corollary 1, \(\dim(\mathcal{H})=d\) captures the dimension of the quantum system used for statistical inference. The number of utilized qubits, if a qubit arrangement is used to create this space, is \(\lceil \log_2(d)\rceil\). If \(d<N\), the upper bound in Corollary 1 is unnecessarily reduced by the dimension of the quantum system. The tightness of Corollary 1 implies that there exists, at least, one inference problem for which the performance can be improved by increasing the dimension of the underlying quantum system. This points to that the minimum number of qubits required for accurately solving a generic inference problem must be above \(\log_2(N)\). We are not asserting that \(\log_2(N)\) is the optimal number of required qubits, but that this is a lower bound for how many qubits are needed to solve the most ‘complicated’ inference problems effectively (in the sense that the inference quality cannot be improved by increasing the dimension of the underlying Hilbert space). Interestingly, the only thing that matters, in this observation, is the size of the support set of the input \(X\) (not its distribution, not the output \(Z\), not the quantum computing method used, and not the classical post-processing procedure implemented). Therefore, this observation is rather universal.

4 Optimal Universal Encoding↩︎

The upper bound in Theorem 1, which is a function of the maximal quantum leakage \(\mathcal{Q}(X\rightarrow A)_{\rho}\), only depends on the quantum encoding of the classical data denoted by \(\mathcal{R}\). This bound is also tight in the sense that it is saturated for at least one inference task. Therefore, maximizing \(\mathcal{Q}(X\rightarrow A)_{\rho}\) provides a good universal encoding policy. This encoder can unlock the barrier in achieving a high accuracy in quantum-assisted statistical inference by increasing the upper bound in Theorem 1. Maximizing the upper bound in ?? , via maximizing \(\mathcal{Q}(X\rightarrow A)_{\rho}\), does not make the bound looser as this bound is always attained for at least one inference problem. The universal optimal encoder is given by \[\begin{align} \label{eqn:max95encoding} \mathop{\mathrm{arg\,max}}_{\rho^x\in \mathcal{S}(\mathcal{H}),\forall x\in\mathcal{X}} \mathcal{Q}(X\!\to\!A)_\rho. \end{align}\tag{2}\] In the next proposition, we prove that this optimization problem attains its maximum over pure states. This is an important revelation as most quantum computing platforms and procedures rely on pure states.

Note that Proposition 3 does not claim that the solution is unique. The problem 2 might admit several solutions but at least one of those solutions involves pure states for encoding classical data. All the optimal solutions have the same maximal quantum leakage.

When considering pure state encodings \(\mathcal{R}=\{\rho^x\}_{x\in\mathcal{X}}\) with \(\rho^x=\ket{\psi_x}\bra{\psi_x}\), \(\forall x\in\mathcal{X}\), with slight abuse of notation, we refer to \(\{\ket{\psi_x}\}_{x\in\mathcal{X}}\) as the state encoding. We consider several examples achieving the optimal encoding.

Basis encoding, also called index encoding [20], is thus not merely a practical convenience but the provably optimal universal encoder whenever the Hilbert space is large enough to accommodate orthogonal codewords. The interesting and more subtle case is \(N>d\), to which we now turn. The following lemma transforms the problem of finding an optimal encoding to an algebraic condition.

Phase encoding is not the only tight frame. Therefore, the optimal encoding is not unique. Among all tight frames achieving \(\mathcal{Q}(X\to A)_{\rho}=\log (d)\), some are more symmetric than others. In what follows, we consider pairwise similarity between codewords and present symmetric constructions when possible.

5 Equiangular Tight Frames↩︎

5.1 The Welch bound↩︎

The natural measure of pairwise similarity between codewords \(\ket{\psi_x}\) and \(\ket{\psi_{x'}}\), \(x\neq x'\), is the squared overlap \(|\bra{\psi_x}\ket{\psi_{x'}}|^2\), which is known as the pure state fidelity in quantum information theory [11]. The following theorem gives a fundamental lower bound on the worst-case overlap. The proof is based on the seminal work of Welch in information theory in [22] and is presented here for the sake of completeness.

Equality in Theorem 6 holds if and only if all pairwise squared overlaps are equal: \(|\langle\psi_x|\psi_{x'}\rangle|^2=c^2:=(N-d)/[d(N-1)]\) for all \(x\neq x'\). This motivates the following definition.

ETFs are optimal quantum encodings in two complementary senses. They achieve the maximum leakage \(\mathcal{Q}(X\to A)_\rho=\log (d)\) (tight unit-norm frame condition), and they simultaneously minimize the maximum pairwise overlap among all tight frames (Welch bound saturation). A small coherence \(c_{N,d}\) means the codewords are as “spread out” as possible in the Hilbert space \(\mathcal{H}\cong\mathbb{C}^d\), which in turn means they are maximally distinguishable in a minimax sense.

Table 1 summarises the key special cases of ETF encodings. When \(N=d\) the ETF degenerates to an orthonormal basis (the complete regime of Proposition 4). As \(N\) increases past \(d\), the coherence \(c_{N,d}\) increases from \(0\) (orthogonal codewords). We now provide explicit codeword sets for these parameter regimes. Existence of an ETF\((N,d)\) is not guaranteed for all \((N,d)\); the problem is connected to deep questions in combinatorics and algebraic number theory [23], [24].

5.2 Regular simplex: \(N=d+1\)↩︎

The regular simplex ETF exists for every \(d\geq1\) and corresponds to \(d+1\) equidistant points on the unit sphere in \(\mathbb{C}^d\), i.e., the vertices of a regular simplex inscribed in the sphere.

5.3 SIC-POVMs via the Heisenberg–Weyl group (\(N=d^2\))↩︎

SIC-POVMs are ETFs\((d^2,d)\) with coherence \(1/\sqrt{d+1}\). A constructive approach in arbitrary dimension uses the discrete Heisenberg–Weyl group.

The significance of SIC-POVMs as optimal encodings is two-fold. First, they saturate the Welch bound with the highest possible coherence \(c_{d^2,d}=1/\sqrt{d+1}\). Second, the self-referential measurement \(M_{p,q}^*=(1/d)\ket{\psi_{p,q}}\bra{\psi_{p,q}}\) is simultaneously optimal for discrimination and has the property that the POVM elements are proportional to the codewords, which is a remarkable self-duality that underpins the role of SIC-POVMs in quantum tomography [27].

6 Numerical Experiments↩︎

6.1 Investigating Popular Quantum Encoding Policies↩︎

We survey seven encoding strategies that span the space from well-known quantum computing encodings to information-theoretically optimal constructions.

1. Basis encoding: For \(x = 0,\ldots,N-1\), the encoded state is \(\rho^x=\ket{x}\bra{x}\), where \(\{\ket{0},\ldots,\ket{d-1}\}\) is the standard computational basis of \(\mathcal{H}\cong\mathbb{C}^d\). This code is valid only if \(N\leq d\). For \(N>d\), codewords must cycle through the basis modulo \(d\), which result in severe information loss. By Proposition 4, basis encoding is universally optimal in the complete regime of \(N\leq d\). Basis encoding is the standard binary representation used in quantum algorithms appearing in Grover’s algorithm, quantum phase estimation, and the HHL algorithm.

2. Phase (DFT) encoding: The classical value \(x = 0,\ldots,N-1\) is encoded as a phase twist \(e^{2\pi ixj/N}\) applied to the uniform superposition resulting in state encoding \(\rho^x=\ket{\psi_x}\bra{\psi_x}\) with \(\ket{\psi_x}=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1} e^{2\pi i xj/N}\,\ket{j}\). Each codeword is a column of a generalized DFT matrix of size \(d\times N\). The frame operator is \(S=\frac{N}{d}\,I_d\) whenever \(d\leq N\). Hence phase encoding is a tight frame for every \(N\geq d\), achieving \(\mathcal{Q}(X\to A)_\rho=\log (d)\) in the over-complete regime. This means phase encoding is the optimal universal encoding for all \(N\geq d\). However, phase encoding is generally not equiangular, that is, the pairwise overlaps \(|\bra{\psi_x}\ket{\psi_{x'}}| = |d^{-1}\sum_j e^{2\pi i(x-x')j/N}|\) depend on \(x-x'\) and grow toward \(1\) as \(N\) increases. Phase encoding is the basis of the quantum phase estimation circuit and the quantum Fourier transform. The codewords \(\ket{\psi_x}\) are eigen-states of the shift operator, making phase encoding the natural representation for periodic signals.

3. Amplitude encoding: Following the standard quantum machine learning convention [37], amplitude encoding represents a classical vector \(\mathbf{v}\in\mathbb{R}^d\) as \(\ket{\psi_{\mathbf{v}}} = \|\mathbf{v}\|^{-1}\sum_j v_j\ket{j}\). Applied to a class label \(x\in\{0,\ldots,N-1\}\), we extract a binary feature vector with a leading bias bit, \(\mathbf{b}(x) = (1,\;\mathrm{bit}_{n-1}(x),\;\ldots,\;\mathrm{bit}_0(x)) \in\{0,1\}^{n+1}\), \(n = \lceil\log_2 N\rceil\), where \(\mathrm{bit}_k(x)\) is the \(k\)-th bit of \(x\) (most significant bit first) and the leading \(1\) ensures \(\mathbf{b}(x)\neq\mathbf{0}\) for all \(x\). The codeword is then \(\ket{\psi_x}= \|\mathbf{b}(x)\|^{-1}\sum_j b_{j}(x)\ket{j}\). The natural dimension is \(d = n+1 = \lceil\log_2 N\rceil + 1\), giving an exponential compression, i.e., \(N\) labels in \(O(\log N)\) dimensions. The frame operator is generally not proportional to identity and the encoding is not a tight frame. Consequently \(\mathcal{Q}(X\to A)_\rho<\log (d)\) in general. However, unlike basis encoding, it uses only \(d=O(\log N)\) dimensions, exploiting the exponential compression of amplitude encoding. In practice, preparing amplitude-encoded states requires \(O(N)\) gates without QRAM [37], which may offset the dimensional compression for large \(N\).

4. Equatorial encoding: For \(x = 0,\ldots,N-1\), the encoded state is \(\rho^x=\ket{\psi_x}\bra{\psi_x}\), where \(\ket{\psi_x}= \cos\!(\frac{\pi x}{N})\ket{0} + \sin\!(\frac{\pi x}{N})\ket{1}\). The angle \(\pi x/N\in[0,\pi)\) sweeps a half-circle on the Bloch great circle, placing codewords at equal angular spacing \(\pi/N\). For \(d=2\) (qubit), the \(N\) codewords span \(\mathbb{C}^2\) for all \(N\geq 2\), and the frame operator evaluates to \(S=(N/2)I_2\), confirming a tight frame for every \(N\). This encoding therefore achieves \(\mathcal{Q}(X\to A)_\rho=\log (2)\) for all \(N\geq 2\), \(d=2\). For \(d>2\) the codewords all lie in the two-dimensional subspace \(\operatorname{span}\{\ket{0},\ket{1}\}\) and do not span \(\mathcal{H}\cong\mathbb{C}^d\). The qubit trine (\(N=3\)) is special cases of equatorial encoding.

5. Dense angle encoding: For \(x = 0,\ldots,N-1\), the encoded state is \(\rho^x=\ket{\psi_x}\bra{\psi_x}\), where \(\ket{\psi_x} = \cos\!(\frac{2\pi x}{N})\ket{0} + \sin\!(\frac{2\pi x}{N})\ket{1}\). The angle \(2\pi x/N\) sweeps a full circle on the Bloch great circle. The doubled angle relative to equatorial encoding means adjacent codewords are \(2\pi/N\) apart (the same angular spacing as DFT columns on the unit circle). For \(N=2\) the codewords are \(\ket{0}\) and \(-\ket{0}\), identical density matrices, so \(P_{\rm guess}(\{1/N,\rho^x\})=1/2\) (random guessing) and \(\mathcal{Q}(X\to A)_\rho=0\). This is the worst possible case. For \(N\geq 3\) the codewords span \(\mathbb{C}^2\) and the frame operator again gives a tight frame for \(d=2\), recovering \(\mathcal{Q}(X\to A)_\rho=\log (2)\). As with equatorial encoding, the subspace restriction to \(\{\ket{0},\ket{1}\}\) makes performance suboptimal for \(d>2\). Dense angle (or “double angle”) encoding appears in quantum kernel methods [38] where the feature map \(\phi(x)=\cos(2x)Z+\sin(2x)Y\) is applied to a qubit initialized in \(\ket{+}\). The doubled angle is motivated by the desire to use the full Bloch sphere range, but the inference analysis shows this does not improve \(\mathcal{Q}(X\to A)_\rho\) over equatorial encoding on a qubit and therefore can suffer for some inference problem.

6. Hamiltonian encoding: Let \(H=\operatorname{diag}(h_0,\ldots,h_{d-1})\) with \(h_k = 2k/(d-1)-1\in[-1,1]\) equally spaced eigenvalues, and let \(\ket{+} = d^{-1/2}\mathbf{1}\) be the uniform superposition. The Hamiltonian encoding is \(\rho^x=\ket{\psi_x}\bra{\psi_x}\), where \(\ket{\psi_x}= e^{-iHt_x}|{+}\rangle = \frac{1}{\sqrt{d}}\sum_{k=0}^{d-1} e^{-ih_k \pi x/N}\,|k\rangle\) with \(t_x = \frac{\pi x}{N}\). The classical value \(x\) is encoded as an evolution time under the Hamiltonian \(H\). Since \(e^{-iHt_x}\) is a diagonal unitary, Hamiltonian encoding is a special case of phase encoding with phases \(e^{-ih_k\pi x/N}\) replacing \(e^{2\pi ixk/N}\). For \(d=2\) with \(H=Z=\operatorname{diag}(1,-1)\), the codewords are \(\ket{\psi_x} = \frac{1}{\sqrt{2}}( e^{-i\pi x/N}\ket{0}+ e^{+i\pi x/N}\ket{1}),\) which form a tight frame for every \(N\), giving \(\mathcal{Q}(X\to A)_\rho=\log (2)\). For \(d>2\) the non-uniform phase structure (\(h_k\) are not integer multiples of \(2\pi/N\)) breaks perfect frame tightness in general, so \(\mathcal{Q}(X\to A)_\rho\) is slightly below \(\log (d)\), but the gap decreases as \(N\) grows. Hamiltonian encoding arises naturally in quantum simulation and quantum sensing, where the signal \(x\) may represent a physical parameter (field strength, coupling constant) that drives a Hamiltonian \(H(x)=xH_0\) for some fixed \(H_0\). After a fixed evolution time the probe state \(e^{-iH(x)t}\ket{+}\) encodes \(x\) as a phase. Lloyd et al. [39] propose Hamiltonian simulation as a feature map for quantum-enhanced machine learning, showing that random Hamiltonians can generate kernels that are hard to evaluate classically.

7. Random encoding: For \(x = 0,\ldots,N-1\), the encoded state is \(\rho^x=\ket{\psi_x}\bra{\psi_x}\), where \(\ket{\psi_x}= v_x/\|v_x\|\) with \(v_x = u_x^{(1)} + i\,u_x^{(2)}\) such that \(u_x^{(1)},u_x^{(2)}\) are independent real Gaussian vectors with mean zero and unit variance. This produces states distributed according to the Haar measure on the unit sphere \(S^{2d-1}\subset\mathbb{C}^d\). By concentration of measure on the sphere, the frame operator of \(N\) i.i.d.Haar-random unit vectors concentrates around \((N/d)I_d\) for large \(N\). More precisely, for fixed \(d\) and \(N\to\infty\), \(S/N \to I_d/d\) almost surely (law of large numbers on the sphere), so random encodings approach tight frames asymptotically.

Lemma 2 proves that tight frames are optimal encodings as they attain \(\mathcal{Q}(X\to A)=\log (d)\) for \(N\geq d\). Phase, equatorial (qubit), and Hamiltonian encodings are tight frames for \(d=2\) at all \(N\). For \(d>2\) only Phase and the explicit ETF constructions (trine, SIC-POVM, simplex) maintain this property. Amplitude (the variant discussed above), equatorial, and dense-angle encodings for \(d>2\) confine all codewords to the two-dimensional \(\{\ket{0},\ket{1}\}\) subspace. They fail the spanning condition of Lemma 2 and achieve \(\mathcal{Q}(X\to A)_\rho\leq\log (2)<\log (d)\). Among all tight frames achieving \(\mathcal{Q}(X\to A)_\rho=\log (d)\), ETFs uniquely satisfy the Welch bound. Phase encoding achieves \(\mathcal{Q}(X\to A)_\rho=\log (d)\) but with coherence far exceeding the Welch bound while the trine, SIC-POVM, and simplex achieve both in their corresponding feasible parameter regions. Finally, random encoding is a strong practical solution. For large \(N/d\), random states concentrate near tight frames and achieve near-optimal \(\mathcal{Q}(X\to A)_\rho\) without any design effort. Explicit ETFs are preferred only when the Welch-bound coherence constraint (e.g.for quantum key distribution or tomography) matters.

Figure 5 reports the fraction of the theoretical maximum leakage achieved by each encoding \(\mathcal{Q}(X\!\to\!A)_\rho / \log(d)\) across fifteen \((N,d)\) parameter pairs and five encoding common encoding policies with the three optimal ETF constructions shown below the separator for reference. Several patterns are immediately apparent. Phase (DFT) encoding achieves the maximum \(\mathcal{Q}(X\!\to\!A)_\rho / \log(d) = 1.00\) in every cell, confirming that it forms a tight frame for all \(N\geq d\) regardless of dimension. Equatorial encoding matches this performance for \(d=2\) across all \(N\), but degrades as \(d\) increases because all codewords lie in the two-dimensional subspace \(\mathrm{span}\{\ket{0},\ket{1}\}\) and fail to span \(\mathbb{C}^d\) when \(d>2\) violating the spanning condition. Amplitude encoding is applicable only when \(d \geq \lceil\log_2 N\rceil + 1\), and achieves substantially suboptimal leakage but improving as \(N/d\) grows. Random Haar encoding approaches optimality for large \(N/d\), consistent with the concentration-of-measure argument that Haar-random states approximate tight frames asymptotically, yet never exactly attains the ceiling. Basis encoding achieves optimality, however, it is only applicable when \(N\leq d\). The three ETF rows confirm the paper’s main results on their optimality.

6.2 Classification via Quantum Machine Learning↩︎

We study a \(4\)-class quantum classification task. The input alphabet is \(\mathcal{X}=\{0,\ldots,7\}\) (\(N=8\)), the Hilbert space is \(\mathcal{H}\cong \mathbb{C}^{4}\) (\(d=4\), two qubits). The output label \(Z\in\{0,1,2,3\}\) is drawn from a fixed balanced random partition of \(\mathcal{X}\) (two tokens per class), chosen once and held constant across all encodings and seeds. With a uniform input prior, \(\max_z P\{Z=z\}=\tfrac{1}{4}\).

The variational quantum circuit (VQC) consists of \(L\in\{1,2,4,8,16\}\) layers, each comprising per-qubit \((R_Y\)–\(R_Z)\) rotations followed by a CNOT gate and per-qubit depolarizing noise with rate \(p=0.01\) per qubit per layer, matching a realistic near-term device. The same ansatz is used with every encoding so that differences in classification accuracy are attributable solely to the encoding itself. State preparation is also subject to the same depolarizing channel. Parameters are trained with the Adam optimizer (learning rate \(0.05\)) using exact parameter-shift gradients and the cross-entropy loss, evaluated on all \(N=8\) input-output pairs per step. Results are averaged over 10 independent random initializations. Figure 6 illustrates the variational quantum circuit.

We compare four encodings of phase (DFT), random Haar, amplitude, and equatorial encoding. The exact maximal quantum leakage \(Q(X\to A)_\rho=\log(N \cdot P_{\rm guess}(\{1/N,\rho^x\}_{x\in\mathcal{X}}))\) is computed for each encoding via a semi-definite program, with results reported in Table 2. As expected, phase encoding achieves the theoretical maximum \(Q(X\to A)_\rho=\log (d) = 2\) bits, with random Haar within 5% of this ceiling, consistent with the concentration-of-measure argument.

Figure 7 plots classification accuracy against circuit depth \(L\). The results stratify into three tiers that map precisely onto the maximal quantum leakage value ranking in Table 2. In Tier 1, phase encoding sits as an exact tight frame attaining maximal quantum leakage, with random Haar as an approximate tight frame attaining near-maximal leakage. Both encodings reach near-perfect or perfect classification by \(L=8\). Phase encoding achieves \(100\%\) accuracy at \(L=16\) while random Haar reaches \(87.5\%\)–\(100\%\). In Tier 2, amplitude encoding sits as an example of a partial frame. Accuracy grows from \(\approx 31\%\) at \(L=1\) to \(\approx 56\%\) at \(L=16\), yet plateaus well below Tier 1. No amount of additional circuit depth bridges this gap. Binary amplitude encoding achieves the exponential dimensional compression promised by amplitude encoding, but it sacrifices leakage relative to tight-frame alternatives. At the bottom, in Tier 3, equatorial sits as an example of encoding that wastes the quantum system’s potential by only confining the encoding to two dimensions. Equatorial encoding degrades toward the random baseline at large depth, as the circuit overfits a fundamentally two-dimensional measurement structure. Figure 8 shows epoch-by-epoch accuracy at fixed depth \(L=4\). Amplitude converges to an intermediate plateau while phase encoding climbs steeply to near-perfect performance and equatorial collapses toward the random baseline.

7 Conclusions↩︎

We have developed and validated a complete information-theoretic theory of optimal universal quantum encoding for statistical inference. This is done by rigorously establishing that quantum maximal leakage is the figure of merit for measuring quality of quantum encoding of classical data. Following this, we can compute the optimal encoding by maximizing maximal quantum leakage. The central result is an exact two-regime characterization of the optimal encoder. In the complete regime (\(N \leq d\)), basis encoding achieves the absolute maximum \(Q(X\to A)_\rho = \log (N)\). However, in the overcomplete regime (\(N > d\)), the maximum leakage \(Q(X\to A)_\rho = \log (d)\) is achieved if the codewords form a tight frame with phase encoding being one such encoding. Given tight frames are not unique, we focus on symmetric choices by investigating equiangular tight frames (ETFs) as the uniquely symmetric optimal encodings. The numerical experiments corroborate the theory across two complementary programmes. The leakage heatmap confirms that phase encoding attains largest maximal quantum leakage for every parameter choice when \(N\geq d\), while random Haar encoding approaches this ceiling asymptotically. We also use a classification experiment to reveal a clean performance hierarchy aligned with the maximal quantum leakage value ranking, with tight-frame encodings (phase exactly, random Haar asymptotically) performing best.

Two directions for future work are particularly natural. First, the present theory assumes ideal state preparation. Extending the optimal characterization to noisy channels would replace the tight-frame condition with a channel-dependent analogue and is directly relevant to the noisy intermediate-scale quantum (NISQ) settings studied experimentally. Second, the optimal encodings identified here, e.g., ETFs, SIC-POVMs, but tight frames more broadly, may require state-preparation circuits of substantial depth or non-Clifford gate count, costs that are prohibitive on near-term hardware. A resource-aware theory of quantum encoding would characterize the best trade-off between maximal quantum leakage and the minimum circuit complexity required to realize the codewords, measured in, for example, two-qubit gate count, \(T\)-gate count in the fault-tolerant setting, or entanglement cost. Such a framework would transform the present information-theoretic optimality conditions into practically actionable design principles, making explicit the price in quantum resources that must be paid for each additional bit of leakage and, conversely, identifying the cheapest encoding that meets a prescribed leakage target.

References↩︎

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1. An earlier version of this paper was presented at 2024 Quantum Techniques in Machine Learning (QTML) at the University of Melbourne.↩︎