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Simulating quantum computers with probabilistic methods

Maarten Van den Nest

Published Jan 1, 2016• 57 citations
Publisher

Abstract

The study of quantum computations that can be simulated efficiently classically is of interest for numerous reasons. From a fundamental point of view, such an investigation sheds light on the intrinsic computational power harnessed in quantum mechanics as compared to classical physics. More practically, understanding which quantum computations do not offer any speed-up over classical computation provides insights in where (not) to look for novel quantum algorithmic primitives. In this talk we discuss novel classical simulation methods that are centered on probabilistic methods ('weak simulation'). We show how these techniques outperform existing methods that rely on the exact computation of measurement probabilities ('strong simulation'). Using weak simulation methods, several new classes of simulatable quantum circuits are generated. For example, we show that various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform an! d other circuits are classically simulatable. Finally, we focus on famous quantum algorithms and investigate the origin of their computational power, or their lack thereof.

DOI
10.14288/1.0103166

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The study of quantum computations that can be simulated efficiently classically is of interest for numerous reasons. From a fundamental point of view, such an investigation sheds light on the intrinsic computational power harnessed in quantum mechanics as compared to classical physics. More practically, understanding which quantum computations do not offer any speed-up over classical computation provides insights in where (not) to look for novel quantum algorithmic primitives. In this talk we discuss novel classical simulation methods that are centered on probabilistic methods ('weak simulation'). We show how these techniques outperform existing methods that rely on the exact computation of measurement probabilities ('strong simulation'). Using weak simulation methods, several new classes of simulatable quantum circuits are generated. For example, we show that various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform an! d other circuits are classically simulatable. Finally, we focus on famous quantum algorithms and investigate the origin of their computational power, or their lack thereof.
We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove that sparse quantum circuits as well as circuits composed of CNOT and exp⁡[iθX]\exp[{i\theta X}]exp[iθX] gates can be simulated classically. In a second part, we apply our results to the simulation of quantum algorithms. It is shown that a recent quantum algorithm, concerned with the estimation of Potts model partition functions, can be simulated efficiently classically. Finally, we show that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes. Specifically, we prove that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.
We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove that sparse quantum circuits as well as circuits composed of CNOT and exp[iaX] gates can be simulated classically. In a second part, we apply our results to the simulation of quantum algorithms. It is shown that a recent quantum algorithm, concerned with the estimation of Potts model partition functions, can be simulated efficiently classically. Finally, we show that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes. Specifically, we prove that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.
The study of quantum computations that can be simulated efficiently classically is of interest for numerous reasons. On a fundamental level, such an investigation sheds light on the intrinsic computational power that is harnessed in quantum mechanics as compared to classical physics. More practically, understanding which quantum computations do not offer any speed-ups over classical computation provides insights in where (not) to look for novel quantum algorithmic primitives. On the other hand, classical simulation of many-body systems is a challenging task, as the dimension of the Hilbert space scales with the number of particles. Therefore, to understand the properties of the systems, suitable approximation methods need to be employed. The lectures will be divided into two parts. In the first part we discuss classical simulation of quantum computation from several perspectives. We review a number of well-known examples of classically simulatable quantum computations, such as the Gottesman-Knill theorem, matchgate simulation and tensor contraction methods. We further discuss simulation methods that are centred on classical sampling methods (‘weak simulation’), and illustrate how these techniques outperform methods that rely on the exact computation of measurement probabilities (‘strong simulation’). The second part focuses on "Entanglement and variational wavefunctions in quantum many body physics". We review the idea of entanglement in quantum many-body systems and how it helps us to understand the success of numerical renormalization group methods. In particular we will discuss a few variational wave-function based methods for simulating strongly correlated quantum systems, which include (1) matrix product states (2) multiscale entanglement renormalization ansatz (3) projected entangled pair states and (4) continuous matrix product states for quantum field theories.
The study of quantum computations that can be simulated efficiently classically is of interest for numerous reasons. On a fundamental level, such an investigation sheds light on the intrinsic computational power that is harnessed in quantum mechanics as compared to classical physics. More practically, understanding which quantum computations do not offer any speed-ups over classical computation provides insights in where (not) to look for novel quantum algorithmic primitives. On the other hand, classical simulation of many-body systems is a challenging task, as the dimension of the Hilbert space scales with the number of particles. Therefore, to understand the properties of the systems, suitable approximation methods need to be employed. The lectures will be divided into two parts. In the first part we discuss classical simulation of quantum computation from several perspectives. We review a number of well-known examples of classically simulatable quantum computations, such as the Gottesman-Knill theorem, matchgate simulation and tensor contraction methods. We further discuss simulation methods that are centred on classical sampling methods (‘weak simulation’), and illustrate how these techniques outperform methods that rely on the exact computation of measurement probabilities (‘strong simulation’). The second part focuses on "Entanglement and variational wavefunctions in quantum many body physics". We review the idea of entanglement in quantum many-body systems and how it helps us to understand the success of numerical renormalization group methods. In particular we will discuss a few variational wave-function based methods for simulating strongly correlated quantum systems, which include (1) matrix product states (2) multiscale entanglement renormalization ansatz (3) projected entangled pair states and (4) continuous matrix product states for quantum field theories.
This paper presents a novel approach to simulating quantum computing algorithms using Monte Carlo methods, specifically designed to account for the inherent probabilistic nature of quantum systems. Traditional simulation methods often rely on deterministic models, which fail to capture the fundamental uncertainty associated with quantum mechanics. Our method employs a stochastic simulation engine that accurately models the probabilistic behavior of quantum gates and measurements. The core of the approach involves generating a large number of samples, each representing a possible outcome of a quantum computation. By analyzing the statistical distribution of these samples, we can approximate the expected results of the algorithm. This technique offers a practical way to evaluate the performance of quantum algorithms, especially those with high levels of uncertainty, and provides a valuable tool for algorithm design and optimization. The method avoids the limitations of purely deterministic simulations and offers a more realistic representation of quantum computation. Key aspects include the development of a robust sampling strategy and the efficient calculation of probabilities. The approach is demonstrated through a detailed explanation and illustrative examples, focusing on the conceptual framework and the underlying mathematical principles.
We study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. The normal form highlights why Clifford circuits have such limited computational power in spite of their high entangling power. At the same time, the normal form shows how the classical simulation of Clifford circuits fits into the standard way of embedding classical computation into the quantum circuit model. This leads to simple extensions of Clifford circuits which are classically simulatable. These circuits can be efficiently simulated by classical sampling (``weak simulation'') even though the problem of exactly computing the outcomes of measurements for these circuits (``strong simulation'') is proved to be #P\#\mathbf{P}#P-complete---thus showing that there is a separation between weak and strong classical simulation of quantum computation.