We show that any language solvable in nondeterministic time exp( exp(⋯exp(n))), where the number of iterated exponentials is an arbitrary function R(n), can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness 1 and soundness 1 − exp(−Cexp(⋯exp(n))), where the number of iterated exponentials is R(n)−1 and C>0 is a universal constant. The result was previously known for R=1 and R=2; we obtain it for any time-constructible function R.

Clifford circuits - i.e. circuits composed of only CNOT, Hadamard, and pi/4 phase gates - play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) - where one additionally conjugates every qubit by the same one-qubit gate U - can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary U can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture. This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets.

We explore the space "just above" BQP by defining a complexity class naCQP (non-adaptive Collapse-free Quantum Polynomial time) which is larger than BQP but does not contain NP relative to an oracle. The class is defined by imagining that quantum computers can perform (non-adaptive) measurements that do not collapse the wavefunction. This non-physical model of computation can efficiently solve problems such as Graph Isomorphism and Approximate Shortest Vector which are believed to be intractable for quantum computers. Furthermore, it can search an unstructured N-element list in Õ(N1/3) time, but no faster than Ω(N1/4), and hence cannot solve NP-hard problems in a black box manner. In short, this model of computation is more powerful than standard quantum computation, but only slightly so. This is surprising as most modifications of BQP increase the power of quantum computation to NP or beyond.

We give a quantum interactive proof system for the local Hamiltonian problem on n qubits in which (i) the verifier has a single round of interaction with five entangled provers, (ii) the verifier sends a classical message on O(log n) bits to each prover, who replies with a constant number of qubits, and (iii) completeness and soundness are separated by an inverse polynomial in $n$. As the same class of proof systems, without entanglement between the provers, is included in QCMA, our result provides the first indication that quantum multiprover interactive proof systems with entangled provers may be strictly more powerful than unentangled-prover interactive proof systems. A distinguishing feature of our protocol is that the completeness property requires honest provers to share a large entangled state, obtained as the encoding of the ground state of the local Hamiltonian via an error-correcting code. Our result can be interpreted as a first step towards a multiprover variant of the quantum PCP conjecture.

We present a protocol which allows a client to have a server carry out a quantum computation for her such that the client's inputs, outputs and computation remain perfectly private, and where she does not require any quantum computational power or memory. The client only needs to be able to prepare single qubits randomly chosen from a finite set and send them to the server, who has the balance of the required quantum computational resources. Our protocol is interactive: after the initial preparation of quantum states, the client and server use two-way classical communication which enables the client to drive the computation, giving single-qubit measurement instructions to the server, depending on previous measurement outcomes. Our protocol works for inputs and outputs that are either classical or quantum. We give an authentication protocol that allows the client to detect an interfering server; our scheme can also be made fault-tolerant. We also generalize our result to the setting of a purely classical client who communicates classically with two non-communicating entangled servers, in order to perform a blind quantum computation. By incorporating the authentication protocol, we show that any problem in BQP has an entangled two-prover interactive proof with a purely classical verifier. Our protocol is the first universal scheme which detects a cheating server, as well as the first protocol which does not require any quantum computation whatsoever on the client's side. The novelty of our approach is in using the unique features of measurement-based quantum computing which allows us to clearly distinguish between the quantum and classical aspects of a quantum computation.