Nonadaptive fault-tolerant verification of quantum supremacy with noise

Abstract

Quantum samplers are believed capable of sampling efficiently from distributions that are classically hard to sample from. We consider a sampler inspired by the classical Ising model. It is nonadaptive and therefore experimentally amenable. Under a plausible conjecture, classical sampling upto additive errors from this model is known to be hard. We present a trap-based verification scheme for quantum supremacy that only requires the verifier to prepare single-qubit states. The verification is done on the same model as the original sampler, a square lattice, with only a constant overhead. We next revamp our verification scheme in two distinct ways using fault tolerance that preserves the nonadaptivity. The first has a lower overhead based on error correction with the same threshold as universal quantum computation. The second has a higher overhead but an improved threshold (1.97%) based on error detection. We show that classically sampling upto additive errors is likely hard in both these schemes. Our results are applicable to other sampling problems such as the Instantaneous Quantum Polynomial-time (IQP) computation model. They should also assist near-term attempts at experimentally demonstrating quantum supremacy and guide long-term ones.

Accrediting outputs of noisy intermediate-scale quantum computing devices

“…In the first two theorems the number ε is replaced by the soundness cr e (see definition 4 in appendix D.2), namely the probability that Alice accepts a wrong output for the target when Bob is cheating. Our mesothetic verification protocol is different from prepare-and-send [26][27][28][29][30][31][32][33] or receive-and-measure [34][35][36][37][38] cryptographic protocols in that Alice encrypts the computation through the QOTP during the actual implementation of the circuits, and not before or after the implementation. To do this, she must possess an nqubit quantum memory and be able to execute single-qubit gates.…”

“…In these protocols (which we call 'cryptographic protocols') the outputs of the target circuit are verified through an interaction between a trusted verifier and an untrusted prover ( figure 1(a)). The verifier is typically allowed to possess a noiseless quantum device able to prepare [26][27][28][29][30][31][32][33]…”

“…Inspired by cryptographic protocols [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44], our accreditation protocol is trap-based, meaning that the target circuit being accredited is implemented together with a number v of classically simulable circuits (the 'trap' circuits) able to detect all types of noise subject to conditions N1 and N2 above. A single run of our protocol requires implementing the target circuit being accredited and v trap circuits.…”

“…Suppose that this protocol is run d times with i.i.d. noise (a standard assumption in trap-based cryptographic protocols [29,36]) and suppose that N 0 acc > protocol runs end with flag bit in the state accñ | . For each of these N acc runs, the state of the system at the end of the protocol run is thus of the form (see equation (3)…”

“…For instance, suppose that the target circuit contains a few hundred qubits and a few hundred gates. Cryptographic protocols require implementing this circuit on a large cluster state containing thousands of qubits and entangling gates [26][27][28][29][30][31]34] or on two spatially-separated devices sharing thousands of copies of Bell states [40,41]; or appending several teleportation gadgets to the target circuit (one for each T-gate in the circuit and six for each Hadamard gate) [33]; or building Feynman-Kitaev clock states, which require entangling the system with an auxiliary qubit per gate in the target circuit [35,39,43]. These protocols are scalable, as they require a number of additional qubits, gates and measurements growing linearly with the size of the target circuit, yet they remain impractical for NISQ devices.In this paper we present an accreditation protocol that provides an upper-bound on the variation distance between noisy and noiseless probability distribution of the outputs of a NISQ device, under the assumption that N1: Noise in state preparation, entangling gates, and measurements is an arbitrary Completely Positive Trace Preserving (CPTP) map encompassing the whole system and the environment (equation (2));…”