Unbiased simulation of near-Clifford quantum circuits

Modeling and simulation are essential for predicting and verifying the behavior of fabricated quantum circuits, but existing simulation methods are either impractically costly or require an unrealistic simplification of error processes. We present a method of simulating noisy Clifford circuits that is both accurate and practical in experimentally relevant regimes. In particular, the cost is weakly exponential in the size and the degree of non-Cliffordness of the circuit. Our approach is based on the construction of exact representations of quantum channels as quasiprobability distributions over stabilizer operations, which are then sampled, simulated, and weighted to yield unbiased statistical estimates of circuit outputs and other observables. As a demonstration of these techniques, we simulate a Steane [[7,1,3]]-encoded logical operation with non-Clifford errors and compute its fault tolerance error threshold. We expect that the method presented here will enable studies of much larger and more realistic quantum circuits than was previously possible.

Figure 1

Monte Carlo simulation of a qubit undergoing repeated coherent rotations, as revealed by its projection onto the +1 eigenstate of the Pauli $Y$ operator. Dashed black line: The analytical result $\langle Y^{+}\rangle =(1+sin\theta )/2$. Solid red line: Estimate using an exact stabilizer decomposition of the rotation channel. Dash-dot blue line: Estimate using an approximate positive decomposition. Shaded bands cover the area one standard deviation above and below the estimates.

Figure 2

Block structure of the circuit simulated as demonstration of our simulation method. The circuit prepares a reference logical state in the Steane [[7,1,3]] encoding. The encoded state is then subjected to errors and, optionally, a round of noisy error correction. The logical error of the resulting state is obtained by removing correctable errors with a round of error-free error correction and then projecting the result onto the originally encoded state. Details of the subcircuits, including specific gates and locations of error insertion, are provided in Appendix pp2.

Figure 3

(Solid blue line with shaded uncertainty band) Estimated infidelity of a Steane [[7,1,3]] logical identity operation as a function of physical qubit depolarization. The circuit implementing the logical operation is specified by Fig. 2. For comparison, the physical infidelity is shown by the dashed black line.

Figure 4

Solid blue line with shaded uncertainty band: Estimated infidelity of a Steane [[7,1,3]] logical identity operation as a function of physical qubit amplitude damping. The circuit implementing the logical operation is specified by Fig. 2. For comparison, the physical infidelity is shown by the dashed black line.

Figure 5

Quantum circuit for encoding Steane [[7,1,3]] quantum error correction codeword.

Figure 6

Quantum circuit for logical identity operation, referred to as a no-op.

Figure 7

Quantum circuit for extracting syndrome 1.

Figure 8

Quantum circuit for extracting syndrome 2.

Figure 9

Quantum circuit for extracting syndrome 3.

Figure 10

Quantum circuit for extracting syndrome 4.

Figure 11

Quantum circuit for extracting syndrome 5.

Figure 12

Quantum circuit for extracting syndrome 6.