Verification of Many-Qubit States

Verification is a task to check whether a given quantum state is close to an ideal state or not. In this paper, we show that a variety of many-qubit quantum states can be verified with only sequential single-qubit measurements of Pauli operators. First, we introduce a protocol for verifying ground states of Hamiltonians. We next explain how to verify quantum states generated by a certain class of quantum circuits. We finally propose an adaptive test of stabilizers that enables the verification of all polynomial-time-generated hypergraph states, which include output states of the Bremner-Montanaro-Shepherd-type instantaneous quantum polynomial time (IQP) circuits. Importantly, we do not make any assumption that the identically and independently distributed copies of the same states are given: Our protocols work even if some highly complicated entanglement is created among copies in any artificial way. As applications, we consider the verification of the quantum computational supremacy demonstration with IQP models, and verifiable blind quantum computing.Comment: 15 pages, 3 figures, published versio

Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur.Comment: 30 pages, 4 figure

Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements

Measurement-based quantum computing is one of the most promising quantum computing models. Although various universal resource states have been proposed so far, it was open whether only two Pauli bases are enough for both of universal measurement-based quantum computing and its verification. In this paper, we construct a universal hypergraph state that only requires $X$ and $Z$-basis measurements for universal measurement-based quantum computing. We also show that universal measurement-based quantum computing on our hypergraph state can be verified in polynomial time using only $X$ and $Z$-basis measurements. Furthermore, in order to demonstrate an advantage of our hypergraph state, we construct a verifiable blind quantum computing protocol that requires only $X$ and $Z$-basis measurements for the client.Comment: 12 pages, 8 figures, 1 table, close to published versio

Sumcheck-based delegation of quantum computing to rational server

Delegated quantum computing enables a client with a weak computational power to delegate quantum computing to a remote quantum server in such a way that the integrity of the server is efficiently verified by the client. Recently, a new model of delegated quantum computing has been proposed, namely, rational delegated quantum computing. In this model, after the client interacts with the server, the client pays a reward to the server. The rational server sends messages that maximize the expected value of the reward. It is known that the classical client can delegate universal quantum computing to the rational quantum server in one round. In this paper, we propose novel one-round rational delegated quantum computing protocols by generalizing the classical rational sumcheck protocol. The construction of the previous rational protocols depends on gate sets, while our sumcheck technique can be easily realized with any local gate set. Furthermore, as with the previous protocols, our reward function satisfies natural requirements. We also discuss the reward gap. Simply speaking, the reward gap is a minimum loss on the expected value of the server's reward incurred by the server's behavior that makes the client accept an incorrect answer. Although our sumcheck-based protocols have only exponentially small reward gaps as with the previous protocols, we show that a constant reward gap can be achieved if two non-communicating but entangled rational servers are allowed. We also discuss that a single rational server is sufficient under the (widely-believed) assumption that the learning-with-errors problem is hard for polynomial-time quantum computing. Apart from these results, we show, under a certain condition, the equivalence between $rational$ and $ordinary$ delegated quantum computing protocols. Based on this equivalence, we give a reward-gap amplification method.Comment: 28 pages, 1 figure, Because of the character limitation, the abstract was shortened compared with the PDF fil

Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups

In this paper we consider what can be computed by a user interacting with a potentially malicious server, when the server performs polynomial-time quantum computation but the user can only perform polynomial-time classical (i.e., non-quantum) computation. Understanding the computational power of this model, which corresponds to polynomial-time quantum computation that can be efficiently verified classically, is a well-known open problem in quantum computing. Our result shows that computing the order of a solvable group, which is one of the most general problems for which quantum computing exhibits an exponential speed-up with respect to classical computing, can be realized in this model

Divide-and-conquer verification method for noisy intermediate-scale quantum computation

Several noisy intermediate-scale quantum computations can be regarded as logarithmic-depth quantum circuits on a sparse quantum computing chip, where two-qubit gates can be directly applied on only some pairs of qubits. In this paper, we propose a method to efficiently verify such noisy intermediate-scale quantum computation. To this end, we first characterize small-scale quantum operations with respect to the diamond norm. Then by using these characterized quantum operations, we estimate the fidelity $\langle\psi_t|\hat{\rho}_{\rm out}|\psi_t\rangle$ between an actual $n$-qubit output state $\hat{\rho}_{\rm out}$ obtained from the noisy intermediate-scale quantum computation and the ideal output state (i.e., the target state) $|\psi_t\rangle$. Although the direct fidelity estimation method requires $O(2^n)$ copies of $\hat{\rho}_{\rm out}$ on average, our method requires only $O(D^32^{12D})$ copies even in the worst case, where $D$ is the denseness of $|\psi_t\rangle$. For logarithmic-depth quantum circuits on a sparse chip, $D$ is at most $O(\log{n})$, and thus $O(D^32^{12D})$ is a polynomial in $n$. By using the IBM Manila 5-qubit chip, we also perform a proof-of-principle experiment to observe the practical performance of our method.Comment: 17 pages, 7 figures, v3: Added a proof-of-principle experiment (Sec. IV) and improved Sec. V, Accepted for publication in Quantu