It was recently shown that for reasonable notions of approximation of states and functions by quantum circuits, almost all states and functions are exponentially hard to approximate [Knill 1995]. The bounds obtained are asymptotically tight except for the one based on total variation distance (TVD). TVD is the most relevant metric for the performance of a quantum circuit. In this paper we obtain asymptotically tight bounds for TVD. We show that in a natural sense, almost all states are hard to approximate to within a TVD of 2/e-εeven for exponentially small ε. The quantity 2/e is asymptotically the average distance to the uniform distribution. Almost all states with probability amplitudes concentrated in a small fraction of the space are hard to approximate to within a TVD of 2-ε. These results imply that non-uniform quantum circuit complexity is non-trivial in any reasonable model. They also reinforce the notion that the relative information distance between states (which is based on the difficulty of transforming one state to another) fully reflects the dimensionality of the space of qubits, not the number of qubits.
It was recently shown that for reasonable notions of approximation of states and functions by quantum circuits, almost all states and functions are exponentially hard to approximate [Knill 1995]. The bounds obtained are asymptotically tight except for the one based on total variation distance (TVD). TVD is the most relevant metric for the performance of a quantum circuit. In this paper we obtain asymptotically tight bounds for TVD. We show that in a natural sense, almost all states are hard to approximate to within a TVD of 2/e-εeven for exponentially small ε. The quantity 2/e is asymptotically the average distance to the uniform distribution. Almost all states with probability amplitudes concentrated in a small fraction of the space are hard to approximate to within a TVD of 2-ε. These results imply that non-uniform quantum circuit complexity is non-trivial in any reasonable model. They also reinforce the notion that the relative information distance between states (which is based on the difficulty of transforming one state to another) fully reflects the dimensionality of the space of qubits, not the number of qubits.
Random quantum circuits aim to efficiently reproduce the statistical properties of ideal random quantum evolution. One approach is to construct approximate unitary designs, which match the moments of Haar-random unitaries up to a prescribed order with controlled error. In this work, we establish quantitative guarantees for how accurately these designs reproduce the full distributions of individual output probabilities. We show that, for every strong ε-approximate unitary k-design on n qubits, the distribution of each individual output probability is within O(k(k+2n)2n+ε) in Kolmogorov distance of the finite-dimensional Porter-Thomas distribution, a beta distribution with parameters 1 and 2n−1. This bound is optimal up to constant factors, implying that a substantially better uniform bound requires additional structure. We further show that local invariance yields an exponential improvement in the dependence on k, with guarantees also in the stronger metric of total variation: every strong ε-approximate unitary k-design invariant under unitary translations acting on logk+O(1) qubits achieves error 2−Ω(k)+O(ε) in Kolmogorov distance and 2−Ω(k)+O(εlog(2/ε)) in total variation distance. These results identify both the distributional accuracy guaranteed by moment matching and the additional structure that substantially improves it. As an application, we show that if the marginal distribution of each output probability of a strong approximate design is within Kolmogorov distance η of its Haar counterpart, then the expected Shannon entropy of the output distribution differs from its Haar value by O(η).
We introduce distance measures between quantum states, measurements, and channels based on their statistical distinguishability in generic experiments. Specifically, we analyze the average Total Variation Distance (TVD) between output statistics of protocols in which quantum objects are intertwined with random circuits and measured in standard basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects – the average-case distances (ACDs). We apply them to analyze the effects of noise in quantum advantage experiments and for efficient discrimination of high-dimensional states and channels without quantum memory. We argue that ACDs are better suited for assessing the quality of NISQ devices than common distance measures such as trace distance or the diamond norm.
In this article, we analyzed the performance of several classes of quantum circuits under various errors. We executed NISQ circuits on both noisy simulator and real hardware backend publicly available from IBM Quantum. On simulation, we created noise models with both individual errors and combination of multiple errors to quantify how each type of error impact performance of quantum circuits. We utilized statistical distances such as total variation distance (TVD), Jensen-Shannon divergence (JSD), and Hellinger distance, to measure the deviation of noisy output from ideal output. Finally, we compare the noisy outputs from the simulator and outputs from the real hardware to showcase the gap between simulation and real device experiment. This gap points to the limitation of present noise models failure to exhibit the behavior of real hardware.
In this article, we analyzed the performance of several classes of quantum circuits under various errors. We executed NISQ circuits on both noisy simulator and real hardware backend publicly available from IBM Quantum. On simulation, we created noise models with both individual errors and combination of multiple errors to quantify how each type of error impact performance of quantum circuits. We utilized statistical distances such as total variation distance (TVD), Jensen-Shannon divergence (JSD), and Hellinger distance, to measure the deviation of noisy output from ideal output. Finally, we compare the noisy outputs from the simulator and outputs from the real hardware to showcase the gap between simulation and real device experiment. This gap points to the limitation of present noise models failure to exhibit the behavior of real hardware.
Abstract Currently available quantum computers, so called Noisy Intermediate-Scale Quantum devices, are characterized by relatively low number of qubits and moderate gate fidelities. In such scenario, the implementation of quantum error correction is impossible and the performance of those devices is quite modest. In particular, the depth of circuits implementable with reasonably high fidelity is limited, and the minimization of circuit depth is required. Such depths depend on the efficiency of the universal set of gates used in computation, and can be bounded using the Solovay–Kitaev theorem. However, it is known that much better, asymptotically tight bounds of the form , can be obtained for specific . Those bounds are controlled by so called spectral gap, denoted . Yet, the computation of is not possible for general and in practice one considers spectral gap at a certain scale r ( ϵ ) , denoted . This turns out to be sufficient to bound the efficiency of provided that one is interested in a physically feasible case, in which an error ϵ is bounded from below. In this paper we derive lower bounds on and, as a consequence, on the efficiency of universal sets of d-dimensional quantum gates satisfying an additional condition. The condition is naturally met for generic quantum gates, such as e.g. Haar random gates. Our bounds are explicit in the sense that all parameters can be determined by numerical calculations on existing computers, at least for small d. This is in contrast with known lower bounds on which involve parameters with ambiguous values.