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
We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in
Applying
- NSF-PAR ID:
- 10378886
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Theory of Computing Systems
- Volume:
- 67
- Issue:
- 1
- ISSN:
- 1432-4350
- Page Range / eLocation ID:
- p. 149-177
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract and the corresponding noiseless output distribution$$p_{\text {noisy}}$$ shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmark$$p_{\text {ideal}}$$ F that measures this correlation behaves as , where$$F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))$$ is the probability of error per circuit location and$$\epsilon $$ 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 and the uniform distribution$$p_{\text {noisy}}$$ decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as$$p_{\text {unif}}$$ . In other words, although at least one local error occurs with probability$$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}$$ , 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$$1-F$$ . Thus, the “white-noise approximation” is meaningful when$$O(F\epsilon \sqrt{s})$$ , a quadratically weaker condition than the$$\epsilon \sqrt{s} \ll 1$$ requirement to maintain high fidelity. The bound applies if the circuit size satisfies$$\epsilon s\ll 1$$ , which corresponds to only$$s \ge \Omega (n\log (n))$$ logarithmic depth circuits, and if, additionally, the inverse error rate satisfies , which is needed to ensure errors are scrambled faster than$$\epsilon ^{-1} \ge {\tilde{\Omega }}(n)$$ 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. -
Abstract In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:
Certifying that a list of
n integers has no 3-SUM solution can be done in Merlin–Arthur time . Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n)$$ time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$ and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$ time).$$\tilde{O}(n^{1.5})$$ Counting the number of
k -cliques with total edge weight equal to zero in ann -node graph can be done in Merlin–Arthur time (where$${\tilde{O}}(n^{\lceil k/2\rceil })$$ ). For odd$$k\ge 3$$ k , this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm -edge graph can be done in Merlin–Arthur time . Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only count$${\tilde{O}}(m)$$ k -cliques in unweighted graphs, and had worse running times for smallk .Computing the All-Pairs Shortest Distances matrix for an
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