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arXiv:2405.07137

Oracle Separation between Noisy Quantum Polynomial Time and the Polynomial Hierarchy

Nai-Hui Chia, Min-Hsiu Hsieh, Hung, Shih-Han, En-Jui Kuo

Published May 12, 2024• 0 citations
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Abstract

This work investigates the oracle separation between the physically motivated complexity class of noisy quantum circuits, inspired by definitions such as those presented by Chen, Cotler, Huang, and Li (2022). We establish that with a constant error rate, separation can be achieved in terms of NP. When the error rate is $Ω(\log n/n)$, we can extend this result to the separation of PH. Notably, our oracles, in all separations, do not necessitate error correction schemes or fault tolerance, as all quantum circuits are of constant depth. This indicates that even quantum computers with minor errors, without error correction, may surpass classical complexity classes under various scenarios and assumptions. We also explore various common noise settings and present new classical hardness results, generalizing those found in studies by Raz and Tal (2022) and Bassirian, Bouland, Fefferman, Gunn, and Tal (2021), which are of independent interest.

DOI
10.48550/arxiv.2405.07137

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This work investigates the oracle separation between the physically motivated complexity class of noisy quantum circuits, inspired by definitions such as those presented by Chen, Cotler, Huang, and Li (2022). We establish that with a constant error rate, separation can be achieved in terms of NP. When the error rate is Ω(log⁡n/n)Ω(\log n/n)Ω(logn/n), we can extend this result to the separation of PH. Notably, our oracles, in all separations, do not necessitate error correction schemes or fault tolerance, as all quantum circuits are of constant depth. This indicates that even quantum computers with minor errors, without error correction, may surpass classical complexity classes under various scenarios and assumptions. We also explore various common noise settings and present new classical hardness results, generalizing those found in studies by Raz and Tal (2022) and Bassirian, Bouland, Fefferman, Gunn, and Tal (2021), which are of independent interest.
We study the quantum-classical polynomial hierarchy, QCPH, which is the class of languages solvable by a constant number of alternating classical quantifiers followed by a quantum verifier. Our main result is that QCPH is infinite relative to a random oracle (previously, this was not even known relative to any oracle). We further prove that higher levels of PH are not contained in lower levels of QCPH relative to a random oracle; this is a strengthening of the somewhat recent result that PH is infinite relative to a random oracle (Rossman, Servedio, and Tan 2016). The oracle separation requires lower bounding a certain type of low-depth alternating circuit with some quantum gates. To establish this, we give a new switching lemma for quantum algorithms which may be of independent interest. Our lemma says that for any ddd, if we apply a random restriction to a function fff with quantum query complexity Q(f)≤n1/3\mathrm{Q}(f)\le n^{1/3}Q(f)≤n1/3, the restricted function becomes exponentially close (in terms of ddd) to a depth-ddd decision tree. Our switching lemma works even in a "worst-case" sense, in that only the indices to be restricted are random; the values they are restricted to are chosen adversarially. Moreover, the switching lemma also works for polynomial degree in place of quantum query complexity.
The 222-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating BQP\mathsf{BQP}BQP and PH\mathsf{PH}PH relative to an oracle. A natural question is therefore to ask what are the minimal quantum resources needed to solve this problem. We show that 222-Forrelation can be solved using Instantaneous Quantum Polynomial-time (IQP\mathsf{IQP}IQP) circuits, a restricted model of quantum computation in which all gates commute. Concretely, signed 222-Forrelation can be solved by a classical random choice between two one-query IQP\mathsf{IQP}IQP circuits, while the absolute-value variant uses two independent executions of this randomized procedure. This answers a recent open question of Girish (STOC 2026) on the power of commuting quantum computations. For the Raz-Tal distribution, this randomization is unnecessary. We use this to show that there is an oracle OOO such that IQPO⊈PHO\mathsf{IQP}^O \not\subseteq \mathsf{PH}^OIQPO⊆PHO, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between IQP\mathsf{IQP}IQP and DQC1\mathsf{DQC}_1DQC1​. We prove Fourier growth bounds for multi-query IQP\mathsf{IQP}IQP circuits, including bounds in terms of the size of their accepting set. Our results suggest a possible route toward decision-based quantum advantage within the restricted IQP\mathsf{IQP}IQP model. The key ingredient is an algebraic identity of the quadratic function Q(x)=∑i<jxixjQ(x) = \sum_{i < j} x_ix_jQ(x)=∑i<j​xi​xj​ that allows extracting inner-product phases within an IQP\mathsf{IQP}IQP circuit.
We develop a framework for learning from noisy quantum experiments in which fault-tolerant devices access uncharacterized systems through noisy couplings. Introducing the complexity class NBQP\textsf{NBQP}NBQP ("noisy BQP''), we model noisy fault-tolerant quantum computers that cannot generally error-correct the oracle systems they query. Using this class, we prove that while noise can eliminate the exponential quantum learning advantages of unphysical, noiseless learners, a superpolynomial gap remains between NISQ\textsf{NISQ}NISQ and fault-tolerant devices. Turning to canonical learning tasks in noisy settings, we find that the exponential two-copy advantage for purity testing collapses under local depolarizing noise. Nevertheless, we identify a setting motivated by AdS/CFT in which noise-resilient physical structure restores this quantum learning advantage. We then analyze noisy Pauli shadow tomography, deriving lower bounds characterizing how instance size, quantum memory and noise jointly control sample complexity, and design algorithms with parametrically matching scalings. We study similar tradeoffs in quantum metrology, and show that the Heisenberg-limited sensitivity of existing error-correction-based protocols persists only up to a timescale inverse-polynomial in the error rate per probe qubit. Together, our results demonstrate that the primitives underlying quantum-enhanced experiments are fundamentally fragile to noise, and that realizing meaningful quantum advantages in future experiments will require interfacing noise-robust physical properties with available algorithmic techniques.
We continue the study of the circuit class GC^0, which augments AC^0 with unbounded-fan-in gates that compute arbitrary functions inside a sufficiently small Hamming ball but must be constant outside it. While GC^0 can compute functions requiring exponential-size circuits, Kumar (CCC 2023) showed that switching-lemma lower bounds for AC^0 extend to GC^0 with no loss in parameters. We prove a parallel result for the polynomial method: any lower bound for AC^0[p] obtained via the polynomial method extends to GC^0[p] without loss in parameters. As a consequence, we show that the majority function MAJ requires depth-ddd GC^0[p] circuits of size 2Ω(n1/2(d−1))2^{Ω(n^{1/2(d-1)})}2Ω(n1/2(d−1)), matching the best-known lower bounds for AC^0[p]. This yields the most expressive class of non-monotone circuits for which exponential-size lower bounds are known for an explicit function. We also prove a similar result for the algorithmic method, showing that E^NP requires exponential-size GCC^0 circuits, extending a result of Williams (JACM 2014). Finally, leveraging our improved classical lower bounds, we establish the strongest known unconditional separations between quantum and classical circuit classes. We separate QNC^0 from GC^0 and GC^0[p] in various settings and show that BQLOGTIME is not contained in GC^0. As a consequence, we construct an oracle relative to which BQP lies outside uniform GC^0, extending the Raz-Tal oracle separation between BQP and PH (STOC 2019).
It is not known whether a classical polynomial-time verifier can verify arbitrary polynomial-time quantum computations with an efficient quantum honest prover without computational assumptions. We show that, relative to a classical oracle, such verification is possible for a language beyond the polynomial hierarchy that is decided exactly in quantum polynomial time. The protocol is public-coin, with perfect completeness and soundness error at most 1/4. To our knowledge, no language with such a proof system was previously proved to lie outside AM relative to an oracle. For every unbounded polynomial-time computable function, the oracle can be chosen so that the message count and the oracle query counts of the quantum algorithm, verifier, and honest prover are bounded by this function of the input length. The message count is optimal up to a constant factor, since for a suitable oracle every interactive proof system for the language uses at least a constant fraction of the messages of our protocol on infinitely many lengths, with a constant depending on its verifier. The language is a variant of recursive Fourier sampling in which code tables encode secrets that address data tables. The tables are evaluated recursively, and the oracle stores only the leaves. The quantum algorithm recovers every secret exactly, and the verifier checks the prover's secrets recursively against random code entries. The recursion restricts the input in a correlated way, as the same secret fixes the code bits and chooses the free coordinate, so Håstad's switching lemma does not apply. We prove a switching theorem for these restrictions, which may be of independent interest. For instances of sufficient height, no constant-depth circuit of size 2poly(n)2^{\mathrm{poly}(n)}2poly(n) changes its output with probability above 2−Ω(n)2^{-Ω(n)}2−Ω(n) when the answer of a random instance is flipped by changing a single leaf.
This paper furthers existing evidence that quantum computers are capable of computations beyond classical computers. Specifically, we strengthen the collapse of the polynomial hierarchy to the second level if: (i) Quantum computers with postselection are as powerful as classical computers with postselection (PostBQP = PostBPP), (ii) any one of several quantum sampling experiments (BosonSampling, IQP, DQC1) can be approximately performed by a classical computer (contingent on existing assumptions). This last result implies that if any of these experiment’s hardness conjectures hold, then quantum computers can implement functions classical computers cannot (FBQP≠ FBPP) unless the polynomial hierarchy collapses to its 2nd level. These results are an improvement over previous work which either achieved a collapse to the third level or were concerned with exact sampling, a physically impractical case. The workhorse of these results is a new technical complexity-theoretic result which we believe could have value beyond quantum computation. In particular, we prove that if there exists an equivalence between problems solvable with an exact counting oracle and problems solvable with an approximate counting oracle, then the polynomial hierarchy collapses to its second level, indeed to ZPP^NP.