This paper proposes a general method to estimate the practical average fidelity error using an -approximate unitary 2 t -design. We then build a specific model of local random circuits to construct the -approximate unitary t -design under a tighter bound on the diamond norm of the corresponding error. For a twirled t -tensor product noisy channel constructed by local random circuits, we extend the estimation of average fidelity using approximate unitary 2 t -designs to an ( m + 1 ) -fold noise scenario. Compared to previous randomized benchmarking, the average fidelity estimated by the method of local random circuits requires less experiments but longer circuit lengths for large n quantum system.
This paper proposes a general method to estimate the practical average fidelity error using an -approximate unitary 2 t -design. We then build a specific model of local random circuits to construct the -approximate unitary t -design under a tighter bound on the diamond norm of the corresponding error. For a twirled t -tensor product noisy channel constructed by local random circuits, we extend the estimation of average fidelity using approximate unitary 2 t -designs to an ( m + 1 ) -fold noise scenario. Compared to previous randomized benchmarking, the average fidelity estimated by the method of local random circuits requires less experiments but longer circuit lengths for large n quantum system.
Randomized benchmarking is a useful scheme for evaluation the average fidelity of a noisy quantum circuit. However, it is insensitive to the unitary error. Here, we propose a method of randomized benchmarking in which a unitary t-design is applied and by which the unitary error estimation can be converted to analysis of pseudo-randomness on a set of unitary operators. We give a bound on the number of randomized benchmarking sequences, when performing a unitary t-design on n-qubit d-dimensional system. By applying local random unitary operators, a decomposition of a unitary t-design, the bound is more practical than the previous bound for multi-qubit circuit. We also give a rigorous bound of a diamond norm between arbitrary and uniform distributions of a set of unitary operators to form an ε-approximate unitary t-design. It can be used to quantitatively analyze the corresponding average fidelity and errors in a large-scale quantum circuit.
We propose a method to estimate the average fidelity using the unitary 2 t -design of a twirled noisy channel, which is suitable for large-scale quantum circuits. Compared with the unitary 2-design in randomized benchmarking, the unitary 2 t -design for the twirling of noisy channels is more flexible in construction and can provide more information. In addition, we prove that the proposed method provides an efficient and reliable estimation of the average fidelity in benchmarking multistage quantum gates and estimating the weakly gate- and time-dependent noise. For time-dependent noise, we provide a scheme of moment superoperator to analyze the noise in different experiments. In particular, we give a lower bound on the average fidelity of a channel with imperfect implementation of benchmarking and state preparation and measurement errors (SPAM).
We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. We show that, for local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution pnoisy of a generic noisy circuit instance and the output distribution pideal of the corresponding noiseless instance shrink exponentially with the expected number of gate-level errors, as F=exp(−2sε±O(sε2)), where ε is the probability of error per circuit location and s is the number of two-qubit gates. Furthermore, if the noise is incoherent, the output distribution approaches the uniform distribution punif at precisely the same rate and can be approximated as pnoisy≈Fpideal+(1−F)punif, that is, local errors are scrambled by the random quantum circuit and contribute only white noise (uniform output). Importantly, we upper bound the total variation error (averaged over random circuit instance) in this approximation as O(Fεs), so the "white-noise approximation" is meaningful when εs≪1, a quadratically weaker condition than the εs≪1 requirement to maintain high fidelity. The bound applies when the circuit size satisfies s≥Ω(nlog(n)) and the inverse error rate satisfies ε−1≥Ω~(n). The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; it was an underlying assumption in complexity-theoretic arguments that low-fidelity random quantum circuits cannot be efficiently sampled classically. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.
We give three new algorithms for efficient in-place estimation, without using ancilla qubits, of average fidelity of a quantum logic gate acting on a d-dimensional system using much fewer random bits than what was known so far. Previous approaches for efficient estimation of average gate fidelity replaced Haar random unitaries in the naive estimation algorithm by approximate unitary 2-designs, and sampled them uniformly and independently. In contrast, in our first algorithm we sample the unitaries of the approximate unitary 2-design uniformly using a limited independence pseudorandom generator, a powerful tool from derandomisation theory. This algorithm uses the same number of basic operations as previous efficient algorithms but much fewer number of random bits. Reducing the requirement of classical random bits increases the reliability of estimation as often, high quality random bits are an expensive computational resource. Our second efficient algorithm, based on a 4-quantum tensor product expander, works if the gate dimension d is large. It uses even lesser random bits than the first algorithm, and has the added advantage that it needs to implement only one unitary versus potentially all the unitaries of an approximate 2-design in the first algorithm. Our third efficient algorithm, based on an l-quantum tensor product expander for moderately large values of l, works for all values of the parameters. It uses slightly more random bits than the other algorithms but has the advantage that it needs to implement only a small number of unitaries versus potentially all the unitaries of an approximate 2-design in the first algorithm. This advantage is of great importance to experimental implementations in the near future.
Abstract We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution pnoisy p noisy and the corresponding noiseless output distribution pideal p ideal shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmark F that measures this correlation behaves as F=exp(−2sϵ±O(sϵ2)) F = exp ( - 2 s ϵ ± O ( s ϵ 2 ) ) , where ϵ ϵ is the probability of error per circuit location and s is the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution pnoisy p noisy and the uniform distribution punif p unif decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as pnoisy≈Fpideal+(1−F)punif p noisy ≈ F p ideal + ( 1 - F ) p unif . In other words, although at least one local error occurs with probability 1−F 1 - F , the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by O(Fϵs) O ( F ϵ s ) . Thus, the “white-noise approximation” is meaningful when ϵs≪1 ϵ s ≪ 1 , a quadratically weaker condition than the ϵs≪1 ϵ s ≪ 1
Random unitaries are useful in quantum information and related fields but hard to generate with limited resources. An approximate unitary k-design is a measure over an ensemble of unitaries such that the average is close to a Haar (uniformly) random ensemble up to the first k moments. A strong notion of approximation bounds the distance from Haar randomness in relative error: the weighted twirl induced by an approximate design can be written as a convex combination involving that of an exact design and vice versa. The main focus of our work is on efficient constructions of approximate designs, in particular whether relative-error designs in sublinear depth are possible. We give a positive answer to this question as part of our main results: 1. Twirl-Swap-Twirl: Let A and B be systems of the same size. Consider a protocol that locally applies k-design unitaries to A^k and B^k respectively, then exchanges l qudits between each copy of A and B respectively, then again applies local k-design unitaries. This protocol yields an ε-approximate relative k-design when l = O(k log k + log(1/ε)). In particular, this bound is independent of the size of A and B as long as it is sufficiently large compared to k and 1/ε. 2. Twirl-Crosstwirl: Let A_1, … , A_P be subsystems of a multipartite system A. Consider the following protocol for k copies of A: (1) locally apply a k-design unitary to each A_p for p = 1, … , P; (2) apply a "crosstwirl" k-design unitary across a joint system combining l qudits from each A_p. Assuming each A_p’s dimension is sufficiently large compared to other parameters, one can choose l to be of the form 2 (Pk + 1) log_q k + log_q P + log_q(1/ε) + O(1) to achieve an ε-approximate relative k-design. As an intermediate step, we show that this protocol achieves a k-tensor-product-expander, in which the approximation error is in 2 → 2 norm, using communication logarithmic in k. 3. Recursive Crosstwirl: Consider an m-qudit system with connectivity given by a lattice in spatial dimension D. For every D = 1, 2, …, we give a construction of an ε-approximate relative k-design using unitaries of spatially local circuit depth O ((log m + log(1/ε) + k log k ) k polylog(k)). Moreover, across the boundaries of spatially contiguous sub-regions, unitaries used in the design ensemble require only area law communication up to corrections logarithmic in m. Hence they generate only that much entanglement on any product state input. These constructions use the alternating projection method to analyze overlapping Haar twirls, giving a bound on the convergence speed to the full twirl with respect to the 2-norm. Using von Neumann subalgebra indices to replace system dimension, the 2-norm distance converts to relative error without introducing system size. The Recursive Crosstwirl construction answers one variant of [Harrow and Mehraban, 2023, Open Problem 1], showing that with a specific, layered architecture, random circuits produce relative error k-designs in sublinear depth. Moreover, it addresses [Harrow and Mehraban, 2023, Open Problem 7], showing that structured circuits in spatial dimension D of depth << m^{1/D} may achieve approximate k-designs.