Verifying commuting quantum computations via fidelity estimation of weighted graph states

Abstract

The instantaneous quantum polynomial time (IQP) model is one of promising models to demonstrate a quantum computational advantage over classical computers. If the IQP model can be efficiently simulated by a classical computer, an unlikely consequence in computer science can be obtained (under some unproven conjectures). In order to experimentally demonstrate the advantage using medium or large-scale IQP circuits, it is inevitable to efficiently verify whether the constructed IQP circuits faithfully work. There exist two types of IQP models, each of which is the sampling on hypergraph states or weighted graph states. For the first-type IQP model, polynomial-time verification protocols have already been proposed. In this paper, we propose verification protocols for the secondtype IQP model. To this end, we propose polynomial-time fidelity estimation protocols of weighted graph states for each of the following four situations where a verifier can (i) choose any measurement basis and perform adaptive measurements, (ii) only choose restricted measurement bases and perform adaptive measurements, (iii) choose any measurement basis and only perform non-adaptive measurements, and (iv) only choose restricted measurement bases and only perform non-adaptive measurements. In all of our verification protocols, the verifier's quantum operations are only singlequbit measurements. Since we assume no independent and identically distributed property on quantum states, our protocols work in any situation.

General framework for verifying pure quantum states in the adversarial scenario

“…IX can be applied to verifying any pure state in the adversarial scenario as long as we can construct a verification strategy for the nonadversarial scenario. Here we summarize the applications of this recipe to verifying many important quantum states, some of which have already appeared on arXiv [43,44,48,49]. The main results are shown in Table I.…”

“…Next, consider weighted graph states [57]. Recently, Hayashi and Takeuchi introduced several efficient protocols for verifying the weighted graph state |G associated with any weighted graph G [49]. One of their protocols is based on a coloring of G and adaptive local projective measurements.…”

“…Recently, a powerful approach known as quantum state verification (QSV) has attracted increasing attention [40][41][42]. Efficient verification protocols have been constructed for bipartite pure states [40,41,[43][44][45][46], Greenberger-Horne-Zeilinger (GHZ) states [47], stabilizer states (including graph states) [14-16, 25, 42], hypergraph states [48], weighted graph states [49], and Dicke states [50]. By contrast, the situation is more troublesome when we turn to the adversarial scenario, in which the quantum states of interest are controlled by an untrusted party, Eve.…”

“…Our work is especially helpful to the verification of bipartite pure states [40,41,[43][44][45][46], GHZ states [47], stabilizer states (including graph states) [42,48], hypergraph states [48], weighted graph states [49], and Dicke states [50], for which efficient verification protocols for the nonadversarial scenario have been constructed recently. By virtue of our recipe, all these states can be verified in the adversarial scenario with much higher efficiencies than was possible previously.…”

Efficient Verification of Pure Quantum States in the Adversarial Scenario

“…Recently, a powerful approach known as quantum state verification [5][6][7] has attracted increasing attention. This approach has led to efficient protocols for verifying bipartite pure states [5,6,[8][9][10][11], stabilizer states (including graph states) [7,[12][13][14][15], hypergraph states [16], weighted graph states [17], and Dicke states [18].…”