How many qubits are needed for quantum computational supremacy?

Abstract

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.

Additive-error fine-grained quantum supremacy

“…We thank authors of Ref. [14] for comments on our manuscript, and sharing their draft. We also thank anonymous reviewers, especially one reviewer who told us some errors and an improvement of fine-grained Stockmeyer's theorem.…”

“…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].…”

“…Note: After uploading this paper on arXiv, authors of Ref. [14] told us that they also show independently additive-error fine-grained quantum supremacy results. (Their additive-error results are added in their latest version.)…”

Entanglement scaling in quantum advantage benchmarks

“…Rapid experimental advancements have spawned an international race towards the first experimental quantum adversarial advantage demonstration, in which a quantum computer outperforms a classical one at some task [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. There is likewise interest in understanding the effectiveness of lowdepth quantum circuits for, e.g., machine learning [19] and quantum simulation [14].…”

Entangling an arbitrary pair of qubits in a long ion crystal

“…We are approaching the so-called Noisy Intermediate-Scale Quantum (NISQ) regime [1], where gate error is the limiting factor for the width and depth of a quantum circuit. It is estimated that 50 qubits require more memory to simulate than what modern supercomputers can offer [2], and coherent control with 90 qubits may be sufficient to demonstrate quantum supremacy [3]. Moreover, large quantities of qubits are required to implement certain quantum error correction schemes with high faulttolerant thresholds (physical error rate < 10 −2 ), where each logical qubit typically consists of more than 10 physical qubits [4][5][6].…”