Random Quantum Circuits Transform Local Noise into Global White Noise

Abstract We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution $$p_{\text {noisy}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mtext>noisy</mml:mtext> </mml:msub> </mml:math> and the corresponding noiseless output distribution $$p_{\text {ideal}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mtext>ideal</mml:mtext> </mml:msub> </mml:math> shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmark F that measures this correlation behaves as $$F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:mtext>exp</mml:mtext> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> <mml:mi>ϵ</mml:mi> <mml:mo>±</mml:mo> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> is the probability of error per circuit location and s is the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution $$p_{\text {noisy}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mtext>noisy</mml:mtext> </mml:msub> </mml:math> and the uniform distribution $$p_{\text {unif}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mtext>unif</mml:mtext> </mml:msub> </mml:math> decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as $$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mtext>noisy</mml:mtext> </mml:msub> <mml:mo>≈</mml:mo> <mml:mi>F</mml:mi> <mml:msub> <mml:mi>p</mml:mi> <mml:mtext>ideal</mml:mtext> </mml:msub> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mtext>unif</mml:mtext> </mml:msub> </mml:mrow> </mml:math> . In other words, although at least one local error occurs with probability $$1-F$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> , the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by $$O(F\epsilon \sqrt{s})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mi>ϵ</mml:mi> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Thus, the “white-noise approximation” is meaningful when $$\epsilon \sqrt{s} \ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , a quadratically weaker condition than the $$\epsilon s\ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mi>s</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> requirement to maintain high fidelity. The bound applies if the circuit size satisfies $$s \ge \Omega (n\log (n))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , which corresponds to only logarithmic depth circuits, and if, additionally, the inverse error rate satisfies $$\epsilon ^{-1} \ge {\tilde{\Omega }}(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>≥</mml:mo> <mml:mover> <mml:mi>Ω</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , which is needed to ensure errors are scrambled faster than F decays. The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; for example, it was an underlying assumption in complexity-theoretic arguments that noisy random quantum circuits cannot be efficiently sampled classically, even when the fidelity is low. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.

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