scholarly journals Efficient classical simulation of noisy random quantum circuits in one dimension

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 318 ◽  
Author(s):  
Kyungjoo Noh ◽  
Liang Jiang ◽  
Bill Fefferman

Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.

Author(s):  
Richard Jozsa ◽  
Akimasa Miyake

Let G ( A ,  B ) denote the two-qubit gate that acts as the one-qubit SU (2) gates A and B in the even and odd parity subspaces, respectively, of two qubits. Using a Clifford algebra formalism, we show that arbitrary uniform families of circuits of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines, can be classically efficiently simulated. This reproduces a result originally proved by Valiant using his matchgate formalism, and subsequently related by others to free fermionic physics. We further show that if the n.n. condition is slightly relaxed, to allow the same gates to act only on n.n. and next n.n. qubit lines, then the resulting circuits can efficiently perform universal quantum computation. From this point of view, the gap between efficient classical and quantum computational power is bridged by a very modest use of a seemingly innocuous resource (qubit swapping). We also extend the simulation result above in various ways. In particular, by exploiting properties of Clifford operations in conjunction with the Jordan–Wigner representation of a Clifford algebra, we show how one may generalize the simulation result above to provide further classes of classically efficiently simulatable quantum circuits, which we call Gaussian quantum circuits.


Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 223 ◽  
Author(s):  
Hakop Pashayan ◽  
Stephen D. Bartlett ◽  
David Gross

Investigating the classical simulability of quantum circuits provides a promising avenue towards understanding the computational power of quantum systems. Whether a class of quantum circuits can be efficiently simulated with a probabilistic classical computer, or is provably hard to simulate, depends quite critically on the precise notion of ``classical simulation'' and in particular on the required accuracy. We argue that a notion of classical simulation, which we call EPSILON-simulation (or ϵ-simulation for short), captures the essence of possessing ``equivalent computational power'' as the quantum system it simulates: It is statistically impossible to distinguish an agent with access to an ϵ-simulator from one possessing the simulated quantum system. We relate ϵ-simulation to various alternative notions of simulation predominantly focusing on a simulator we call a poly-box. A poly-box outputs 1/poly precision additive estimates of Born probabilities and marginals. This notion of simulation has gained prominence through a number of recent simulability results. Accepting some plausible computational theoretic assumptions, we show that ϵ-simulation is strictly stronger than a poly-box by showing that IQP circuits and unconditioned magic-state injected Clifford circuits are both hard to ϵ-simulate and yet admit a poly-box. In contrast, we also show that these two notions are equivalent under an additional assumption on the sparsity of the output distribution (poly-sparsity).


Author(s):  
Maximilian Moll ◽  
Leonhard Kunczik

AbstractIn recent history, reinforcement learning (RL) proved its capability by solving complex decision problems by mastering several games. Increased computational power and the advances in approximation with neural networks (NN) paved the path to RL’s successful applications. Even though RL can tackle more complex problems nowadays, it still relies on computational power and runtime. Quantum computing promises to solve these issues by its capability to encode information and the potential quadratic speedup in runtime. We compare tabular Q-learning and Q-learning using either a quantum or a classical approximation architecture on the frozen lake problem. Furthermore, the three algorithms are analyzed in terms of iterations until convergence to the optimal behavior, memory usage, and runtime. Within the paper, NNs are utilized for approximation in the classical domain, while in the quantum domain variational quantum circuits, as a quantum hybrid approximation method, have been used. Our simulations show that a quantum approximator is beneficial in terms of memory usage and provides a better sample complexity than NNs; however, it still lacks the computational speed to be competitive.


2022 ◽  
Vol 18 (1) ◽  
pp. 1-26
Author(s):  
Mario Simoni ◽  
Giovanni Amedeo Cirillo ◽  
Giovanna Turvani ◽  
Mariagrazia Graziano ◽  
Maurizio Zamboni

Classical simulation of Noisy Intermediate Scale Quantum computers is a crucial task for testing the expected performance of real hardware. The standard approach, based on solving Schrödinger and Lindblad equations, is demanding when scaling the number of qubits in terms of both execution time and memory. In this article, attempts in defining compact models for the simulation of quantum hardware are proposed, ensuring results close to those obtained with standard formalism. Molecular Nuclear Magnetic Resonance quantum hardware is the target technology, where three non-ideality phenomena—common to other quantum technologies—are taken into account: decoherence, off-resonance qubit evolution, and undesired qubit-qubit residual interaction. A model for each non-ideality phenomenon is embedded into a MATLAB simulation infrastructure of noisy quantum computers. The accuracy of the models is tested on a benchmark of quantum circuits, in the expected operating ranges of quantum hardware. The corresponding outcomes are compared with those obtained via numeric integration of the Schrödinger equation and the Qiskit’s QASMSimulator. The achieved results give evidence that this work is a step forward towards the definition of compact models able to provide fast results close to those obtained with the traditional physical simulation strategies, thus paving the way for their integration into a classical simulator of quantum computers.


2017 ◽  
Vol 17 (3&4) ◽  
pp. 262-282
Author(s):  
Dax E. Koh

Extended Clifford circuits straddle the boundary between classical and quantum computational power. Whether such circuits are efficiently classically simulable seems to depend delicately on the ingredients of the circuits. While some combinations of ingredients lead to efficiently classically simulable circuits, other combinations, which might just be slightly different, lead to circuits which are likely not. We extend the results of Jozsa and Van den Nest [Quant. Info. Comput. 14, 633 (2014)] by studying two further extensions of Clifford circuits. First, we consider how the classical simulation complexity changes when we allow for more general measurements. Second, we investigate different notions of what it means to ‘classically simulate’ a quantum circuit. These further extensions give us 24 new combinations of ingredients compared to Jozsa and Van den Nest, and we give a complete classification of their classical simulation complexities. Our results provide more examples where seemingly modest changes to the ingredients of Clifford circuits lead to “large” changes in the classical simulation complexities of the circuits, and also include new examples of extended Clifford circuits that exhibit “quantum supremacy”, in the sense that it is not possible to efficiently classically sample from the output distributions of such circuits, unless the polynomial hierarchy collapses.


2018 ◽  
Vol 175 ◽  
pp. 03002
Author(s):  
Joshua R. McKenney ◽  
William J. Porter ◽  
Joaquín E. Drut

Following up on a recent analysis of two cold atoms in a time-dependent harmonic trap in one dimension, we explore the entanglement entropy of two and three fermions in the same situation when driven through a parametric resonance. We find that the presence of such a resonance in the two-particle system leaves a clear imprint on the entanglement entropy. We show how the signal is modified by attractive and repulsive contact interactions, and how it remains present for the three-particle system. Additionaly, we extend the work of recent experiments to demonstrate how restricting observation to a limited subsystem gives rise to locally thermal behavior.


2004 ◽  
Vol 132 (11) ◽  
pp. 2708-2713
Author(s):  
Abdessamad Qaddouri ◽  
Jean Côté

Abstract A direct elliptic boundary value problem solver used for meteorological applications has been optimized. The problem to be solved is symmetric under a parity operation, and this is preserved by discretization. Therefore if the mesh possesses this symmetry, then the discretized problem will share this symmetry as well. The direct method can make use of this symmetry on a variable mesh to reduce the cost associated with the slow transform, a matrix product, by half. It is also shown that this can be combined with the Strassen–Winograd algorithm for even better results.


1999 ◽  
Vol 9 (2) ◽  
pp. 3850-3853 ◽  
Author(s):  
B. Ruck ◽  
Yonuk Chong ◽  
R. Dittmann ◽  
A. Engelhardt ◽  
B. Oelze ◽  
...  

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