Explicit Lower Bounds on Strong Quantum Simulation

Huang, Cupjin; Newman, Michael; Szegedy, Márió

doi:10.1109/tit.2020.3004427

Cited by 23 publications

(22 citation statements)

References 30 publications

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“…In [45], they also provide a compelling argument that it would be difficult to show that NSETH implies a version of the conjectures underlying their (and our) analysis. Additionally, in [33], the finegrained lower bounds for simulation from our work and from [34] (which used SETH to rule out exponentialtime algorithms that compute output probabilities of quantum circuits) were extended to be based on other fine-grained assumptions, including the well-studied Orthogonal Vectors, 3-SUM, and All Pairs Shortest Path conjectures [66]. Other models that are universal under post-selection where this method may apply include various kinds of extended Clifford circuits [38,39], and conjugated Clifford circuits [14].…”

Section: Resultsmentioning

confidence: 99%

“…The possible optimality of Ryser's formula is also bolstered by work in [37], where it is unconditionally proven that a monotone circuit requires n(2 n−1 −1) multiplications to compute the permanent, essentially matching the complexity of Ryser's formula. This was recently extended to show similar lower bounds on monotone circuits that estimate output amplitudes of quantum circuits [34]. Of course, per-int-NSETH(1 − δ) for vanishing δ goes further and asserts that computation via Ryser's formula is optimal even with the power of non-determinism.…”

Section: Theorem 2 Assuming Eth There Exists a Constantmentioning

confidence: 90%

“…Accordingly, the lower bound we derive on the simulation time of boson sampling circuits when we take b = 0.999 is essentially tight, matching the runtime of the naive simulation algorithm up to factors logarithmic in the total runtime. Recently, a similar approach was applied to obtain lower bounds on the difficulty of computing output probabilities of quantum circuits based on the SETH conjecture [34]. Our work has the disadvantage of using a less well-studied and possibly stronger conjecture (poly3-NSETH(a)) but the advantage of ruling out classical algorithms for sampling, i.e.…”

Section: Introductionmentioning

confidence: 99%

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How many qubits are needed for quantum computational supremacy?

Dalzell

Harrow

Koh

et al. 2020

Quantum

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Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy (PH) does not collapse, a stronger version of the statement that P≠NP, which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing fine-grained versions of the non-collapse conjecture. Our first two conjectures poly3-NSETH(a) and per-int-NSETH(b) take specific classical counting problems related to the number of zeros of a degree-3 polynomial in n variables over F2 or the permanent of an n×n integer-valued matrix, and assert that any non-deterministic algorithm that solves them requires 2cn time steps, where c∈{a,b}. A third conjecture poly3-ave-SBSETH(a′) asserts a similar statement about average-case algorithms living in the exponential-time version of the complexity class SBP. We analyze evidence for these conjectures and argue that they are plausible when a=1/2, b=0.999 and a′=1/2.Imposing poly3-NSETH(1/2) and per-int-NSETH(0.999), and assuming that the runtime of a hypothetical quantum circuit simulation algorithm would scale linearly with the number of gates/constraints/optical elements, we conclude that Instantaneous Quantum Polynomial-Time (IQP) circuits with 208 qubits and 500 gates, Quantum Approximate Optimization Algorithm (QAOA) circuits with 420 qubits and 500 constraints and boson sampling circuits (i.e. linear optical networks) with 98 photons and 500 optical elements are large enough for the task of producing samples from their output distributions up to constant multiplicative error to be intractable on current technology. Imposing poly3-ave-SBSETH(1/2), we additionally rule out simulations with constant additive error for IQP and QAOA circuits of the same size. Without the assumption of linearly increasing simulation time, we can make analogous statements for circuits with slightly fewer qubits but requiring 104 to 107 gates.

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Section: Resultsmentioning

confidence: 99%

Section: Theorem 2 Assuming Eth There Exists a Constantmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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How many qubits are needed for quantum computational supremacy?

Dalzell

Harrow

Koh

et al. 2020

Quantum

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show abstract

“…Secondly, their experimental realization is hindered by a generally poor tolerance to errors in the implementation, which is compounded by the necessity to implement circuits with relatively large (say, at least √ N for an N × N grid) depth. Combined with the resort to complexity-theoretic assumptions for which there is little guidance in terms of concrete parameter settings (see however [DHKLP18]), this has led to an ongoing race in efficient simulations [CZX+18,HNS18,MFIB18]. Indeed, the proposals operate in a limited computational regime, requiring a machine with, say, at least 50 qubits (to prevent direct clasical simulation) but at most 70 qubits (so that verification can be performed in a reasonable amount of time)-leaving open the question of what to do with a device with more than, say, 100 qubits.…”

Section: Introductionmentioning

confidence: 99%

Trading Locality for Time: Certifiable Randomness from Low-Depth Circuits

Coudron

Stark

Vidick

2021

Commun. Math. Phys.

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The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using a (classical or quantum) circuit of sub-logarithmic depth. This assumption contrasts with the locality assumption used for randomness certification based on Bell inequality violation and more recent proposals for randomness certification based on computational assumptions. Furthermore, to demonstrate randomness generation it is sufficient for a device to sample from the ideal output distribution within constant statistical distance. Our procedure is inspired by recent work of Bravyi et al. (Science 362(6412):308–311, 2018), who introduced a relational problem that can be solved by a constant-depth quantum circuit, but provably cannot be solved by any classical circuit of sub-logarithmic depth. We develop the discovery of Bravyi et al. into a framework for robust randomness expansion. Our results lead to a new proposal for a demonstrated quantum advantage that has some advantages compared to existing proposals. First, our proposal does not rest on any complexity-theoretic conjectures, but relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth. Second, success on our task can be easily verified in classical linear time. Finally, our task is more noise-tolerant than most other existing proposals that can only tolerate multiplicative error, or require additional conjectures from complexity theory; in contrast, we are able to allow a small constant additive error in total variation distance between the sampled and ideal distributions.

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“…The depth-four model [1], the Boson Sampling model [2], the IQP model [3,4], the one-clean qubit model [5,6,7,8,9], the HC1Q model [10], and the random circuit model [11,12,13] are known examples. These results prohibit only polynomial-time classical sampling, but recently, impossibilities of some exponential-time classical simulations have been shown based on classical fine-grained complexity conjectures [14,15,16,17,18,19].…”

Section: Introductionmentioning

confidence: 99%

Additive-error fine-grained quantum supremacy

Morimae

Tamaki

2020

Quantum

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It is known that several sub-universal quantum computing models, such as the IQP model, the Boson sampling model, the one-clean qubit model, and the random circuit model, cannot be classically simulated in polynomial time under certain conjectures in classical complexity theory. Recently, these results have been improved to ``fine-grained" versions where even exponential-time classical simulations are excluded assuming certain classical fine-grained complexity conjectures. All these fine-grained results are, however, about the hardness of strong simulations or multiplicative-error sampling. It was open whether any fine-grained quantum supremacy result can be shown for a more realistic setup, namely, additive-error sampling. In this paper, we show the additive-error fine-grained quantum supremacy (under certain complexity assumptions). As examples, we consider the IQP model, a mixture of the IQP model and log-depth Boolean circuits, and Clifford+T circuits. Similar results should hold for other sub-universal models.

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Explicit Lower Bounds on Strong Quantum Simulation

Cited by 23 publications

References 30 publications

How many qubits are needed for quantum computational supremacy?

How many qubits are needed for quantum computational supremacy?

Trading Locality for Time: Certifiable Randomness from Low-Depth Circuits

Additive-error fine-grained quantum supremacy

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