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arXiv · 2101.12223

Fast estimation of outcome probabilities for quantum circuits

Hakop Pashayan·Oliver Reardon-Smith·Kamil Korzekwa·Stephen D. Bartlett
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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).

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Hakop Pashayan, Oliver Reardon-Smith, Kamil Korzekwa, Stephen D. Bartlett. 2021-01-28. Fast estimation of outcome probabilities for quantum circuits. https://doi.org/10.1103/prxquantum.3.020361

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