Fast estimation of outcome probabilities for quantum circuits (2101.12223v3)

Abstract: We present two classical algorithms for the simulation of universal quantum circuits on $n$ qubits constructed from $c$ instances of Clifford gates and $t$ arbitrary-angle $Z$-rotation gates such as $T$ gates. Our algorithms complement each other by performing best in different parameter regimes. The $\tt{Estimate}$ algorithm produces an additive precision estimate of the Born rule probability of a chosen measurement outcome with the only source of run-time inefficiency being a linear dependence on the stabilizer extent (which scales like $\approx 1.17^t$ for $T$ gates). Our algorithm is state-of-the-art for this task: as an example, in approximately $13$ hours (on a standard desktop computer), we estimated the Born rule probability to within an additive error of $0.03$, for a $50$-qubit, $60$ non-Clifford gate quantum circuit with more than $2000$ Clifford gates. Our second algorithm, $\tt{Compute}$, calculates the probability of a chosen measurement outcome to machine precision with run-time $O(2^{t-r} t)$ where $r$ is an efficiently computable, circuit-specific quantity. With high probability, $r$ is very close to $\min {t, n-w}$ for random circuits with many Clifford gates, where $w$ is the number of measured qubits. $\tt{Compute}$ can be effective in surprisingly challenging parameter regimes, e.g., we can randomly sample Clifford+$T$ circuits with $n=55$, $w=5$, $c=10^5$ and $t=80$ $T$ gates, and then compute the Born rule probability with a run-time consistently less than $10$ minutes using a single core of a standard desktop computer. We provide a C+Python implementation of our algorithms and benchmark them using random circuits, the hidden shift algorithm and the quantum approximate optimization algorithm (QAOA).

• The paper presents the Estimate and Compute algorithms, which enable efficient classical estimation and computation of quantum circuit measurement probabilities.
• It leverages the stabilizer extent—a measure that scales as 1.17^t for T gates—to quantify the impact of non-Clifford operations on simulation complexity.
• The algorithms demonstrate practical feasibility by accurately simulating large circuits and supporting the verification of NISQ devices.

An Analysis of Classical Algorithms for Quantum Circuit Simulations

The paper "Fast estimation of outcome probabilities for quantum circuits" by Hakop Pashayan et al. presents two classical algorithms designed to simulate the outcome probabilities of quantum circuits. These algorithms, coined as $Estimate$ and $Compute$, offer efficient solutions for evaluating quantum circuits constructed from a combination of Clifford gates and non-Clifford gates, specifically $T$ gates. The primary goal of the research is to estimate or compute the probabilities associated with quantum measurements using classical computational resources, which is particularly relevant in the context of simulating and verifying near-term quantum devices.

Quantum Circuit Simulation Challenges

The paper addresses fundamental challenges in quantum computing, specifically the task of classically simulating quantum circuits which inherently demand exponential resources. The inherent difficulty arises from the universality of the quantum circuit represented by the inclusion of arbitrary-angle $Z$-rotation gates and $T$ gates alongside a background of Clifford gates. Such quantum circuits, when naively simulated using classical methods, require resources that scale unfavorably with the quantum system size.

Main Contributions

1. Algorithmic Approach: The $Estimate$ and $Compute$ algorithms are introduced as complementary methods for simulating quantum circuits. The $Estimate$ algorithm focuses on producing estimates of Born rule probabilities with an additive error bound, particularly efficient when fewer qubits are measured, or a high degree of Clifford gates is utilized. The $Compute$ algorithm, alternatively, calculates probabilities to machine precision and is particularly effective when a large number of qubits can be tightly compressed due to the structure of the circuit.
2. Stabilizer Extent: A central component to the effectiveness of the $Estimate$ algorithm is its leverage on the stabilizer extent, defined as a measure that relates to the number of non-stabilizer (non-Clifford) operations. This stabilizer extent scales approximately as $1.17^t$ for $T$ gates, providing a quantifiable measure to appreciate how non-stabilizer components impact the overall run-time and error estimates.
3. Computational Insights: The algorithms show remarkable improvements in performance, often reducing the computational complexity significantly for certain circuit configurations. For instance, in a benchmark with a complex 50-qubit, 60 non-Clifford gate circuit with over 2000 Clifford gates, the $Estimate$ algorithm provided a probability estimate within an error margin of 0.03 in approximately 13 hours on a standard desktop computer—demonstrating practical feasibility in non-trivial scenarios.

Implications and Future Directions

1. Verification of NISQ Devices: The algorithms provide a vital tool for simulating noisy intermediate-scale quantum (NISQ) systems—helping to verify that quantum devices are operating as intended. As such, they complement quantum hardware by providing a means of cross-checking and validation.
2. Scaling and Feasibility: While classical algorithms like $Estimate$ and $Compute$ do not replicate quantum computation in terms of operational efficiency, they bridge a crucial gap by offering feasible simulation methods for quantum circuits that lie at the edge of classical tractability. Scaling these approaches to increasingly complex circuits remains a compelling challenge and area for future research.
3. Broader Quantum-Classical Hybrids: The constructs developed in this paper suggest pathways to developing enhanced hybrid quantum-classical algorithms, where quantum devices handle operations efficiently executed using quantum resources, while classical simulators manage tasks well-suited to classical algorithms.

Conclusion

The research presented highlights significant advancements in the arena of classical simulation for quantum circuits. By introducing efficient simulation algorithms, the paper not only solves practical problems associated with simulating quantum circuits but also provides foundational insights that can drive further optimization and innovations in the intersection of quantum computing and classical computational techniques. As quantum devices continue to progress, such classical simulators remain an integral component of the quantum ecosystem, providing critical support functions ranging from verification to performance assessment.