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101
Improved simulation of stabilizer circuits
 Phys. Rev. Lett
"... The GottesmanKnill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we ..."
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Cited by 65 (6 self)
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The GottesmanKnill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freelyavailable program called CHP (CNOTHadamardPhase), which can handle thousands of qubits easily. • We show that the problem of simulating stabilizer circuits is complete for the classical complexity class ⊕L, which means that stabilizer circuits are probably not even universal for classical computation. • We give efficient algorithms for computing the inner product between two stabilizer states, putting any nqubit stabilizer circuit into a “canonical form ” that requires at most O ( n 2 /log n) gates, and other useful tasks. • We extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of nonstabilizer gates, and circuits acting on general tensorproduct initial states but containing only a limited number of measurements. 1
Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to ..."
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Cited by 41 (16 self)
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We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
Parallel quantum computation and quantum codes.” quantph/9808027
"... Abstract. We propose a definition of QNC, the quantum analog of the efficient parallel class NC. We exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth, including circuits for encoding and decoding standard quantum errorcorrecting codes, ..."
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Cited by 39 (2 self)
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Abstract. We propose a definition of QNC, the quantum analog of the efficient parallel class NC. We exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth, including circuits for encoding and decoding standard quantum errorcorrecting codes, or more generally any circuit consisting of controllednot gates, controlled πshifts, and Hadamard gates. Finally, while we note the Quantum Fourier Transform can be parallelized to linear depth, we conjecture that an even simpler ‘staircase ’ circuit cannot be parallelized to less than linear depth, and might be used to prove that QNC < QP. 1
Quantum Weakest Preconditions
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2005
"... We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stonetype duality between the usual statetransformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum comput ..."
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Cited by 36 (2 self)
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We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stonetype duality between the usual statetransformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming we develop a notion that is directly taken from concepts used in quantum computation. The proof that weakest preconditions exist for completely positive maps follows immediately from the Kraus representation theorem. As an example we give the semantics of Selinger’s language in terms of our weakest preconditions. We also cover some specific situations and exhibit an interesting link with stabilizers.
Improving gatelevel simulation of quantum circuits
 Quantum Information Processing
"... While thousands of experimental physicists and chemists are currently trying to build scalable quantum computers, it appears that simulation of quantum computation will be at least as critical as circuit simulation in classical VLSI design. However, since the work of Richard Feynman in the early 198 ..."
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Cited by 35 (8 self)
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While thousands of experimental physicists and chemists are currently trying to build scalable quantum computers, it appears that simulation of quantum computation will be at least as critical as circuit simulation in classical VLSI design. However, since the work of Richard Feynman in the early 1980s little progress was made in practical quantum simulation. Most researchers focused on polynomialtime simulation of restricted types of quantum circuits that fall short of the full power of quantum computation [7]. Simulating quantum computing devices and useful quantum algorithms on classical hardware now requires excessive computational resources, making many important simulation tasks infeasible. In this work we propose a new technique for gatelevel simulation of quantum circuits which greatly reduces the difficulty and cost of such simulations. The proposed technique is implemented in a simulation tool called the Quantum Information Decision Diagram (QuIDD) and evaluated by simulating Grover’s quantum search algorithm [8]. The backend of our package, QuIDD Pro, is based on Binary Decision Diagrams, wellknown for their ability to efficiently represent many seemingly intractable combinatorial structures. This reliance on a wellestablished area of research allows us to take advantage of existing software for BDD manipulation and achieve unparalleled empirical results for quantum simulation. 1
Simulating quantum computation by contracting tensor networks
 SIAM Journal on Computing
, 2005
"... The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has treewidth d can be simulated deterministically in T O(1) exp[O(d)] time, which, in particular, is polynomial in T if d = O(logT). Am ..."
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Cited by 30 (1 self)
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The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has treewidth d can be simulated deterministically in T O(1) exp[O(d)] time, which, in particular, is polynomial in T if d = O(logT). Among many implications, we show efficient simulations for quantum formulas, defined and studied by Yao (Proceedings of the 34th Annual Symposium on Foundations of Computer Science, 352–361, 1993), and for logdepth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that oneway quantum computation of Raussendorf and Briegel (Physical Review Letters, 86:5188– 5191, 2001), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a smalltreewidth graph.
Quantum Information Theory and the Foundations of Quantum Mechanics
, 2004
"... This thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of ..."
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Cited by 28 (7 self)
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This thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of information is to be firmly distinguished from the technical notions arising in information theory; noun, hence does not refer to a particular or substance. The popular claim ‘Information is Physical ’ is assessed and it is argued that this proposition faces a destructive dilemma. Accordingly, the slogan may not be understood as an ontological claim, but at best, as a methodological one. A novel argument is provided against Dretske’s (1981) attempt to base a semantic notion of information on ideas from information theory. The function of various measures of information content for quantum systems is explored and the applicability of the Shannon information in the quantum context maintained against the challenge of Brukner and Zeilinger (2001). The phenomenon of quantum teleportation is then explored as a case study serving to emphasize the value of
A Quantum Logic Array Microarchitecture: Scalable Quantum Data Movement and Computation
 Proceedings of the 38th International Symposium on Microarchitecture MICRO38
, 2005
"... Recent experimental advances have demonstrated technologies capable of supporting scalable quantum computation. A critical next step is how to put those technologies together into a scalable, faulttolerant system that is also feasible. We propose a Quantum Logic Array (QLA) microarchitecture that f ..."
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Cited by 28 (3 self)
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Recent experimental advances have demonstrated technologies capable of supporting scalable quantum computation. A critical next step is how to put those technologies together into a scalable, faulttolerant system that is also feasible. We propose a Quantum Logic Array (QLA) microarchitecture that forms the foundation of such a system. The QLA focuses on the communication resources necessary to efficiently support faulttolerant computations. We leverage the extensive groundwork in quantum error correction theory and provide analysis that shows that our system is both asymptotically and empirically fault tolerant. Specifically, we use the QLA to implement a hierarchical, arraybased design and a logarithmic expense quantumteleportation communication protocol. Our goal is to overcome the primary scalability challenges of reliability, communication, and quantum resource distribution that plague current proposals for largescale quantum computing. Our work complements recent work by Balenseifer et al [1], which studies the software tool chain necessary to simplify development of quantum applications; here we focus on modeling a fullscale optimized microarchitecture for scalable computing. 1.
Classical simulation of noninteractingfermion quantum circuits
 Phys. Rev. A
"... We show that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant [1] corresponds to a physical model of noninteracting fermions in one dimension. We give an alternative proof of his result using the language of fermions and extend ..."
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Cited by 25 (2 self)
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We show that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant [1] corresponds to a physical model of noninteracting fermions in one dimension. We give an alternative proof of his result using the language of fermions and extend the result to noninteracting fermions with arbitrary pairwise interactions, where gates can be conditioned on outcomes of complete von Neumann measurements in the computational basis on other fermionic modes in the circuit. This last result is in remarkable contrast with the case of noninteracting bosons where universal quantum computation can be achieved by allowing gates to be conditioned on classical bits [2].