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39
Succinct Quantum Proofs for Properties of Finite Groups
 In Proc. IEEE FOCS
, 2000
"... In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite g ..."
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Cited by 83 (3 self)
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In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite groups whose elements are encoded as strings of a given length and whose group operations are performed by a group oracle. We prove that for an arbitrary group oracle there exist succinct (polynomiallength) quantum proofs for the Group NonMembership problem that can be checked with small error in polynomial time on a quantum computer. Classically this is impossibleit is proved that there exists a group oracle relative to which this problem does not have succinct proofs that can be checked classically with bounded error in polynomial time (i.e., the problem is not in MA relative to the group oracle constructed). By considering a certain subproblem of the Group NonMembership problem we obtain a simple proof that there exists an oracle relative to which BQP is not contained in MA. Finally, we show that quantum proofs for nonmembership and classical proofs for various other group properties can be combined to yield succinct quantum proofs for other group properties not having succinct proofs in the classical setting, such as verifying that a number divides the order of a group and verifying that a group is not a simple group.
A lambda calculus for quantum computation
 SIAM Journal of Computing
"... The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine, and continues to be of enormous benefit in the classical theory of computation. We propos ..."
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Cited by 66 (1 self)
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The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine, and continues to be of enormous benefit in the classical theory of computation. We propose that quantum computation, like its classical counterpart, may benefit from a version of the lambda calculus suitable for expressing and reasoning about quantum algorithms. In this paper we develop a quantum lambda calculus as an alternative model of quantum computation, which combines some of the benefits of both the quantum Turing machine and the quantum circuit models. The calculus turns out to be closely related to the linear lambda calculi used in the study of Linear Logic. We set up a computational model and an equational proof system for this calculus, and we argue that it is equivalent to the quantum Turing machine.
Quantum algorithms for algebraic problems
, 2008
"... Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational pro ..."
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Cited by 24 (2 self)
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Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum
An introduction to quantum complexity theory
 Collected Papers on Quantum Computation and Quantum Information Theory
, 2000
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Twoqubit projective measurements are universal for quantum computation
"... Nielsen showed in quantph/0108020 that universal quantum computation can be performed using projective measurements, quantum memory, and preparation of the 0 〉 state. Furthermore, 4qubit measurements are sufficient. Fenner and Zhang showed in quantph/0111077 that 3qubit measurements are suffici ..."
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Nielsen showed in quantph/0108020 that universal quantum computation can be performed using projective measurements, quantum memory, and preparation of the 0 〉 state. Furthermore, 4qubit measurements are sufficient. Fenner and Zhang showed in quantph/0111077 that 3qubit measurements are sufficient. We prove that 2qubit measurements are sufficient, closing the gap between the upper and lower bound of the number of qubits to be measured jointly. We conclude with some open questions. 1 Introduction and previous work Studying the resources required for universal quantum computation is important not only for its realization but also for our theoretical understanding of what makes it so powerful. In the predominant standard quantum circuit model [1], it suffices to prepare the 0 〉 state, to measure individual qubits in the computation basis, and to well approximate any unitary gate. Any unitary gate
Level reduction and the quantum threshold theorem
 PH.D. THESIS, CALTECH, 2007, EPRINT ARXIV:QUANTPH/0703230
, 2007
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Threshold error penalty for faulttolerant quantum computation with nearest neighbor communication
 IEEE Trans. Nanotech
"... Abstract—The error threshold for faulttolerant quantum computation with concatenated encoding of qubits is penalized by internal communication overhead. Many quantum computation proposals rely on nearest neighbor communication, which requires excess gate operations. For a qubit stripe with a width ..."
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Cited by 9 (0 self)
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Abstract—The error threshold for faulttolerant quantum computation with concatenated encoding of qubits is penalized by internal communication overhead. Many quantum computation proposals rely on nearest neighbor communication, which requires excess gate operations. For a qubit stripe with a width of + 1 physical qubits implementing levels of concatenation, we find that the error threshold of 2.1 10 5 without any communication burden is reduced to 1.2 10 7 when gate errors are the dominant source of error. This 175 penalty in error threshold translates to an 13 penalty in the amplitude and timing of gate operation control pulses. Index Terms—Fault tolerance, quantum information. I.
SELFTESTING OF UNIVERSAL AND FAULTTOLERANT SETS OF QUANTUM GATES
, 2007
"... We consider the design of selftesters for quantum gates. A selftester for the gates F 1,...,F m is a procedure that, given any gates G1,...,Gm, decides with high probability if each Gi is close to F i. This decision has to rely only on measuring in the computational basis the effect of iterating ..."
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We consider the design of selftesters for quantum gates. A selftester for the gates F 1,...,F m is a procedure that, given any gates G1,...,Gm, decides with high probability if each Gi is close to F i. This decision has to rely only on measuring in the computational basis the effect of iterating the gates on the classical states. It turns out that, instead of individual gates, we can design only procedures for families of gates. To achieve our goal we borrow some elegant ideas of the theory of program testing: We characterize the gate families by specific properties, develop a theory of robustness for them, and show that they lead to selftesters. In particular we prove that the universal and faulttolerant set of gates consisting of a Hadamard gate, a cNOT gate, and a phase rotation gate of angle π/4 is selftestable.
Representation of Quantum Circuits with Clifford and π/8 Gates
, 806
"... Abstract. In this paper, we introduce the notion of a normal form of one qubit quantum circuits over the basis {H, P, T}, where H, P and T denote the Hadamard, Phase and π/8 gates, respectively. This basis is known as the standard set and its universality has been shown by Boykin et al. [FOCS ’99]. ..."
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Abstract. In this paper, we introduce the notion of a normal form of one qubit quantum circuits over the basis {H, P, T}, where H, P and T denote the Hadamard, Phase and π/8 gates, respectively. This basis is known as the standard set and its universality has been shown by Boykin et al. [FOCS ’99]. Our normal form has several nice properties: (i) Every circuit over this basis can easily be transformed into a normal form, and (ii) Every two normal form circuits compute same unitary matrix if and only if both circuits are identical. We also show that the number of unitary operations that can be represented by a circuit over this basis that contains at most n Tgates is exactly 192 · (3 · 2 n − 2).
THE PHYSICAL CHURCHTURING THESIS AND THE PRINCIPLES OF QUANTUM THEORY
, 2012
"... As was emphasized by Deutsch, quantum computation shatters complexity theory, but is innocuous to computability theory. Yet Nielsen and others have shown how quantum theory as it stands could breach the physical ChurchTuring thesis. We draw a clear line as to when this is the case, in a way that is ..."
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Cited by 5 (2 self)
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As was emphasized by Deutsch, quantum computation shatters complexity theory, but is innocuous to computability theory. Yet Nielsen and others have shown how quantum theory as it stands could breach the physical ChurchTuring thesis. We draw a clear line as to when this is the case, in a way that is inspired by Gandy. Gandy formulates postulates about physics, such as homogeneity of space and time, bounded density and velocity of information — and proves that the physical ChurchTuring thesis is a consequence of these postulates. We provide a quantum version of the theorem. Thus this approach exhibits a formal nontrivial interplay between theoretical physics symmetries and computability assumptions.