What limits the simulation of quantum computers?

Abstract: It is imperative that useful quantum computers be very difficult to simulate classically; otherwise classical computers could be used for the applications envisioned for the quantum ones. Perfect quantum computers are unarguably exponentially difficult to simulate: the classical resources required grow exponentially with the number of qubits $N$ or the depth $D$ of the circuit. Real quantum computing devices, however, are characterized by an exponentially decaying fidelity $\mathcal{F} \sim (1-\epsilon)^{ND}$ with an error rate $\epsilon$ per operation as small as $\approx 1\%$ for current devices. In this work, we demonstrate that real quantum computers can be simulated at a tiny fraction of the cost that would be needed for a perfect quantum computer. Our algorithms compress the representations of quantum wavefunctions using matrix product states (MPS), which capture states with low to moderate entanglement very accurately. This compression introduces a finite error rate $\epsilon$ so that the algorithms closely mimic the behavior of real quantum computing devices. The computing time of our algorithm increases only linearly with $N$ and $D$. We illustrate our algorithms with simulations of random circuits for qubits connected in both one and two dimensional lattices. We find that $\epsilon$ can be decreased at a polynomial cost in computing power down to a minimum error $\epsilon_\infty$. Getting below $\epsilon_\infty$ requires computing resources that increase exponentially with $\epsilon_\infty/\epsilon$. For a two dimensional array of $N=54$ qubits and a circuit with Control-Z gates, error rates better than state-of-the-art devices can be obtained on a laptop in a few hours. For more complex gates such as a swap gate followed by a controlled rotation, the error rate increases by a factor three for similar computing time.

• The paper introduces a simulation method using matrix product states to compress quantum wavefunctions and efficiently mimic noisy quantum systems.
• It demonstrates that classical resources scale linearly with the number of qubits and circuit depth in low- to moderate-entanglement scenarios.
• Numerical results show practical simulation of 54-qubit circuits with a 1-2% error rate per operation, paralleling experimental quantum supremacy findings.

An Analysis of the Classical Simulation Limits of Quantum Computers

The paper "What limits the simulation of quantum computers?" by Zhou, Stoudenmire, and Waintal explores the significant challenge of accurately simulating quantum computers using classical computing systems. The authors rigorously explore the limitations imposed on simulating quantum computers and report on new methodologies that effectively leverage classical resources to simulate quantum tasks feasibly under certain conditions.

The main premise revolves around the exponential difficulty traditionally faced in simulating a perfect quantum computer classically, which stems from the exponential increase in classical resources with the number of qubits $N$ and circuit depth $D$. This intrinsic difficulty forms the basis of recent interest in demonstrating practical quantum supremacy—where quantum devices surpass traditional computation abilities. However, real-world quantum devices entail decoherence and imprecision, leading to lower-than-potential entanglement levels.

The research introduces a classical simulation approach employing matrix product states (MPS) to compress quantum wavefunctions. This approach aptly handles low to moderate entanglement scenarios with high accuracy and mimics the behavior of real quantum computers, introducing an error rate $\epsilon$. Critically, the computational cost scales linearly with $N$ and $D$, much unlike the exponential scaling seen in exact simulations. The analysis reveals that the error per operation for classically simulating bipolar quantum behavior using this method generally hovers around 1-2%, which aligns closely with the operational fidelity in existing quantum devices.

The paper presents substantial numerical results showcasing simulation scenarios with one- and two-dimensional arrays of qubits using practical classical resources. For instance, a linear cost increase allows for simulation of 54-qubit, two-dimensional quantum circuits akin to those in recent experimental quantum supremacy demonstrations. The notably practical simulation duration using basic computational resources underlines the approach's feasibility.

The theoretical implications extend into understanding the genuine computational capabilities of noisy quantum computers amidst contemporary error thresholds, arguing for a reevaluation of how truly "quantum" some computational problems purportedly exceeding classical capacities are in practice, considering the encumbrance of real device error rates. Conclusively, the research suggests that the exploitable state-space of a noisy quantum computer might be far less than the entirety of its Hilbert space, dictated heavily by fidelity limitations rather than solely on qubits and connectivity.

The findings prompt further exploration into the dimensions of quantum entanglement that provide practical computational utility and emphasize enhanced precision and connectivity as prerequisites for genuinely exploiting the computational power of quantum systems. The efficiency and adaptability of the classical simulation mechanism explored here open doors for future investigations into hybrid computing systems that optimize quantum and classical resources to new ends.