Keywords: quantum computing, random circuit sampling

1 Introduction↩︎

Quantum computing is a rapidly evolving field in which tremendous progress has been made in recent years. A critical challenge for scaling up quantum computers is, however, error control and error correction. Errors are typically inflicted by contact with the environment of the quantum device. This external "noise" leads to a loss of coherence in the quantum device so that its fidelity deteriorates.

To benchmark fidelity and measure the performance of a quantum computer, random circuit sampling (RCS) with cross-entropy benchmarking (XEB) has been identified as a suitable task [1]–[3]. RCS with a quantum computer can be divided into three steps: First, one samples a random circuit by classical means. one then runs the circuit on the quantum computer having a pre-defined initial state, which typically is the ground state of the system. Finally, a bit string is read out from the device, which effectively samples the bit string with a probability determined by the result state of the calculation.

In this paper, we develop an alternative approach to RCS. The basic idea is to directly sample a random state of the underlying Hilbert space instead of sampling a random circuit and applying it to an initial state. We show equivalence of both approaches in Sec. 2.1, and we derive appropriate probability density functions for sampling random states in Sec. 2.2. In Sec. 2.3 we present the basic sampling algorithm, which we revise in Sec. 2.4 with respect to large numbers of qubits. Finally, we discuss our results in Sec. 3.

2 Random circuit sampling↩︎

2.1 Uniformly distributed quantum circuits↩︎

Let us consider a system with \(n\) qubits. To establish appropriate sampling routines, we first identify the spaces and the sets from which to sample.

The single-qubit Hilbert space is $ ^2\(, so that the Hilbert space of the system is\)\(\mathcal{H} = ( \mathrm{l}\mathrm{C}^2)^{\otimes n} =\mathrm{l}\mathrm{C}^D \quad (D = 2^n).\)$ Let us further assume a quantum calculation on the system with input \(v_0\). In recent calculations, \(v_0\) typically is the ground state of the system, but we note that for RCS the actual initial state does not matter. However, \(v_0\) is a state in \(\mathcal{H}\), i.e. \(v_0 \in \mathcal{H}\), and \(\| v_0 \| = 1\). A circuit in a quantum device is described by a unitary transform, \(U\), and the corresponding quantum calculation basically applies \(U\) to \(v_0\), so that the (theoretical) output state is \(v = Uv_0\).

The unitary transforms on \(\mathcal{H}\) are given by the unitary group \(U(D)\), which is a subset of \(\mathcal{H}^2 = \mathrm{l}\mathrm{C}^{2D}\). Since \(U(D)\) is compact, there exists a unique norm-one Haar measure, \(\nu\), which serves as the probability measure in RCS. Due to translation-invariance, \(\nu\) describes uniformly distributed circuits. For an observable, \(F\), in RCS, the expectation value is given by \[\label{rcs1} \langle F \rangle = \int\limits_{U(D)} F(Uv_0) \, d\nu .\tag{1}\] Since \(F\) is a function on the state space of \(\mathcal{H}\), we argue in the following that we obtain the equivalent result by sampling uniformly distributed states: \[\label{rcs2} \langle F \rangle = \int\limits_{S(D)} F(v) \, d\mu .\tag{2}\] Here, \(S(D)\) denotes the subset of states in \(\mathcal{H}\), \[S(D) = \{ v \in \mathcal{H}: \, \|v\| = 1 \},\] and \(\mu\) is the norm-one Lebesgue measure on \(S(D)\). We note that $ ^D$ is topologically isomorphic to $ !^{2D}$, and that \(S(D)\) is topologically isomorphic to the unit sphere, \(S_\mathrm{re}(2D)\), in $ !^{2D}\(. In particular, the Lebesgue measure on\)S(D)$ is invariant with respect to the action of \(U(D)\).
Proposition: The Haar measure \(\nu\) uniquely determines \(\mu\), and equations (1 ) and (2 ) are equivalent for an observable in RCS.
Proof: We apply theorem 25 (p.384) of Ref. [4] in the following: \(U(D)\) is a compact group which acts on the compact set \(S(D)\). The group action is transitive, since for all \(v,w \in S(D)\) there exists a unitary transform \(U\) such that \(v = Uw\). For each \(v \in S(D)\) we further define the function \[\phi_v: U(D) \to S(D), \quad \phi_v(U) = Uv.\] These functions are continuous, so that for a compact subset \(K \subset S(D)\), \(\phi_v^{-1}(K)\) is also compact. The group action is therefore proper, and we can apply the theorem. We note that the measure is given by \(\mu = \nu \circ \phi_v^{-1}\) for any \(v \in S(D)\) and that it also coincides with the norm-one Lebesgue measure \(\mu\). \(\rule{5pt}{5pt}\)

2.2 Probability density functions↩︎

In recent quantum calculations, the read-out after a calculation with \(n\) qubits is a bit string of length \(n\). The set of all possible bit strings, \(\{ b \}\), provides a basis of \(\mathcal{H}\), so that a vector \(v \in \mathcal{H}\) is given by \[v = \sum_b v_b b \, , \quad (v_b) \in\mathrm{l}\mathrm{C}^D.\] Each coordinate can further be represented by its real and imaginary parts, \(v_b = x_b + i y_b\), and the set of states is given by the unit sphere \[S_\mathrm{re}(2D) = \left\{ ((x_b,y_b)) \in\mathrm{I}\!\mathrm{R}^{2D}: \, \sum_b (x_b^2 + y_b^2 ) = 1 \right\}.\] In order to sample a random state from \(S_\mathrm{re}(2D)\), we first consider the probability, \(p_{b_1} = | v_{b_1} |^2\), of an output bit string \(b_1\). We note that if we sample or fix \(v_{b_1}\) for a random state \(v\), then the remaining part is a random vector with norm \(\sqrt{1-p_{b_1}}\), \[v^{(b_1)}= \sum_{b\neq b_1} v_b b\, \in\mathrm{l}\mathrm{C}^{D-1} \quad \left( \| v^{(b_1)} \|^2 = 1-p_{b_1} \right).\] This gives rise to a recursive pattern, which we exploit in the following.

The area of a sphere of radius \(r\) in $ !^{2D}\(,\)S_(2D,r)\(, is given by\)\(A(S_\mathrm{re}(2D,r)) = \frac{2 \pi^D}{(D-1)!} R^{2D-1}.\)$ Note that we use the notation \(S_\mathrm{re}(2D) = S_\mathrm{re}(2D,1)\). Moreover, \[2\pi \int\limits_0^1dr \, \frac{ A(S_\mathrm{re}(2(D-1),r))}{A(S_\mathrm{re}(2D))} = \int\limits_0^1 dr \, 2(D-1) r^{2D-3} = 1.\] We interpret \[\tilde{\rho}_D(r) dr = 2(D-1) r^{2D-3} dr\] as the probability density function (PDF) of uniformly distributed states \(v \in S_\mathrm{re}(2D)\) with \(\| v^{(b_1)} \| = r\) for some bit string \(b_1\). Since we are mainly interested in probabilities in subsequent simulations, we switch to the variable to \(p = r^2\) and obtain the PDF \[\rho_D(p) dp = (D-1) p^{D-2} dp.\] We note that this also yields a PDF for \(p_{b_1} = 1-p\), which converges to the Porter-Thomas PDF for large \(D\): \[\begin{align} (D-1) (1-p_{b_1})^{D-2} \, dp_{b_1} & = & \frac{D-1}{D-2} \left( 1- \frac{\bar{p}_{b_1}}{D-2} \right)^{D-2} d\bar{p}_{b_1} \quad ( \bar{p}_{b_1} = p_{b_1} (D-2)) \, , \\ & \approx & e^{-\bar{p}_{b_1}} \, d\bar{p}_{b_1} \\ & = & D e^{-D \tilde{p}_{b_1}} \, d\tilde{p}_{b_1}\, \quad \left( \tilde{p}_{b_1} = \frac{\bar{p}_{b_1}}{D} = \frac{D-2}{D} p_{b_1} \right). \end{align}\] However, we do not study the technical details of this limit here, since we continue working with the exact PDF, \(\rho_D\).

2.3 Sampling algorithm↩︎

The Monte Carlo methods developed in this paper make use of importance sampling. Let \(x \in [0,1]\) be a random variable, let \(f\) be a continuous positive PDF, and let \[x = F(y) = \int\limits_a^y f(z) \, dz \quad (-\infty \leq a \leq y \leq b \leq \infty).\] Since \(f\) is positive, \(F\) is bijective so that \(y = F^{-1}(x)\) is well-defined. Moreover, \[\frac{dx}{dy} = f(y),\] and we obtain for an integrable function (observable), \(g\), \[\langle g \rangle = \int\limits_a^b f(y) g(y) \, dy = \int\limits _0^1 dx \, g(F^{-1}(x)).\] We note that \(dx\) is the image measure of \(f(y)dy\). For the PDF \(f(p) = \rho_D(p)\) we obtain in particular \[F(p_b) = \int\limits_0^{p_b} (D-1) p^{D-2} \, dp = p_b^{D-1} \, , \quad F^{-1}(x) = x^{\frac{1}{D-1}} \, .\]

To sample a random state, we start with sampling the bit-string probabilities, \(p_b\). As mentioned in Sec. 2.2, we can perform this recursively:
Algorithm 1

1. Sample \(x_1 \in [0,1]\), and set \(p_1= 1 - x_1^{\frac{1}{D-1}}\)

2. For \(2 \leq k \leq D-1\): Sample \(x_k \in [0,1]\), and set \[p_k= \left( 1 - \sum_{l=1}^{k-1} p_l \right) \left( 1 - x_k^{\frac{1}{D-k}} \right)\]

3. Set \[p_D= \left( 1 - \sum_{l=1}^{D-1} p_l \right)\]

4. Sample \(D\) phase angles in \([0,2\pi)\)

5. Sample a permutation \(\sigma\) of \((1,2,...,D)\)
   

We note that the last step reflects the fact that for any state \(v = \sum_b v_b b\) and any permutation \(\sigma\), \(v_\sigma = \sum_b v_{\sigma(b)} b\) is also a state. In particular, performing steps 1 to 4 yields an equivalence class of states, \([(v_{\sigma(b)})]\), and in the last step we sample an element in that equivalence class. However, we illustrate this algorithm for a system with 12 qubits in Fig. 1 (a), which is analogous to the example presented in Ref. [3]. The algorithm is implemented in a spreadsheet without step 4, because only probabilities, \(p_b\), are depicted (see also Ref. [5]).

2.4 Sampling states of larger systems↩︎

For larger systems, fully capturing a random state quickly becomes infeasible. The reason is that reading out the result of a quantum computation is a measurement process, which leads to a collapse of the wave function according to the Copenhagen interpretation of quantum mechanics. In particular, reading out the result yields a bit string, \(b\), with probability \(p_b = |v_b|^2\), where \(v_b\) is the corresponding component of the result state \(v = \sum_b v_b b\). The result state \(v\) is therefore only statistically accessible and can only be evaluated using expectation values of observables, which are obtained by repeating the same computation many times and taking statistical averages. An example is shown for component probabilities of a random state in Fig. 2. We therefore sample only one component when simulating larger systems, which is sufficient for comparison with results of recent quantum computations, see Ref. [1], [2], [6], [7].

A benchmark observable in quantum RCS is cross-entropy fidelity, \(F_{\tiny XEB}\), which is calculated as follows: One first samples a random state, \(v\), and by reading the output bit string, \(b\), one samples \(b\) from \(v\) with probability \(p_b\). Determining further \(p^{(sim)}_b\) with an accompanying classical simulation one calculates \[F_{\tiny XEB} = D \left\langle p^{(sim)}_b \right\rangle - 1 \, .\] From a theoretical perspective, the expectation value of this two-step sampling process is given by \[F_{\tiny XEB} = D \!\!\!\int\limits_{S(D)} \sum_b p_b^2 \, d\mu \, - 1\] Note that \(d\mu\) is the PDF for sampling a random state and that \(p_b\) is the discrete probability for sampling \(b\) during read out. Since \(\mu\) is invariant with respect to permutations of indices (which are unitary transforms), the integral reduces to \[F_{\tiny XEB} = D^2 \!\!\!\int\limits_{S(D)} p_{b_0}^2 \, d\mu - 1= D^2 \! \int\limits_0^1 (1-p)^2 \rho_D(p) \, dp \, - 1 = \frac{D-1}{D+1}\, ,\] where \(b_0\) is a fixed but otherwise arbitrary coordinate (bit string).

To achieve numerical stability with increasing number of qubits, we rescale probabilities by a factor of \(D\), i.e. we introduce \(\bar{p} = pD\) in the sampling algorithm, and we rewrite the inverse distribution function as follows \[\bar{p} = D p = D \left( 1 - x^{\frac{1}{D-1}} \right) = D \left( 1 - \exp\left( \frac{\ln{x}}{D-1}\right)\right) = - D \,\, expm1 \left( \frac{\ln{x}}{D-1}\right) \, .\] We note in particular that for \(D \to \infty\) we obtain \[D \left( 1 - \exp\left( \frac{\ln{x}}{D-1}\right)\right) \to \left. - \frac{d}{dt} \exp(t \ln{x}) \right|_{t=0} = - \ln{x} \, ,\] i.e., for large \(D\), we basically sample from the Porter-Thomas PDF, \(\exp(-\bar{p}) d\bar{p}\). The revised algorithm for large systems with \(n\) qubits (\(n \geq 45\)) now reads as follows:
Algorithm 2

1. Sample \(x \in [0,1]\). For \(n \leq 1000\) (\(D = 2^n\)), set \[\bar{p} = - D \,\, expm1 \left( \frac{\ln{x}}{D-1}\right),\] else \(\bar{p} = - \ln{x}\)

2. Sample a random bit string of length \(n\)

3. Repeating steps 1 and 2 \(N\) times, \(F_{\tiny XEB}\) is estimated as \[\bar{F}_{\tiny XEB} = \frac{1}{N}\sum_{k=1}^N \bar{p}_k^2 - 1.\]

Note that convergence of the estimate to its theoretical value is guaranteed by the central limit theorem. We also note that sampling a bit string of length \(n\) is the time-consuming step for large systems, which can however be improved by parallelization.

For test purposes, algorithm 2 was implemented on a PC, and the results are summarized in table 1. Qubit systems with 70, 105, 1000, and 1048576 (\(2^{20}\)) qubits are chosen for test runs.

3 Discussion↩︎

Random circuit sampling has become a well-established task for measuring fidelity and performance in quantum computing. The quantum problem is hard to solve on a classical device since the computational effort increases exponentially with the number of qubits. In particular, running a random circuit on a quantum computer typically yields a highly entangled result state. Due to the high degree of entanglement, the full Hilbert space needs to be considered, so that both memory consumption and run times increase exponentially for classical devices. Therefore, quantum computers can provide results, where the number of qubits makes calculations on classical devices impossible.

However, the result state of a calculation on a quantum computer is only statistically available, since reading the output is a measurement process, which leads to a collapse of the wavefunction. We can in principle only obtain expectation values of observables with quantum computers, which are calculated by repeating the same computation many times and taking statistical averages. In particular, the bit strings read out in quantum RCS are basically sampled with a probability that is given by the probabilities of the components of the result state, i.e., the squared absolute values of its components. We note also that to verify the correct probability distributions, auxiliary classical calculations are usually required.

In this paper, we develop an alternative approach to RCS. Instead of applying a random circuit to an initial state of a qubit system, we directly sample components of a random state. We have shown equivalence of both approaches in Sec. 2.1, which means that the result of applying a random circuit to an initial state of a qubit system is a random state. To perform the task of RCS, quantum calculations (i. e. using random circuits) are therefore not strictly required.

In Sec. 2.2, we derive exact probability density functions for sampling random states, which we use in sampling algorithms in Sec. 2.3 and in Sec. 2.4. Test runs on a PC demonstrate the extremely low computational cost of this approach; see table 1, so that I would argue that the task of RCS can be conveniently performed on classical PCs.

References↩︎