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264
Quantum complexity theory
 in Proc. 25th Annual ACM Symposium on Theory of Computing, ACM
, 1993
"... Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97–117]. This constructi ..."
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Cited by 574 (5 self)
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Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97–117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specified. We prove that O(log T) bits of precision suffice to support a T step computation. This justifies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church–Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a boundederror probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are efficiently decidable (with small errorprobability) on a quantum Turing machine satisfies BPP ⊆ BQP ⊆ P ♯P. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.
Quantum Error Correction Via Codes Over GF(4)
, 1997
"... The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on s ..."
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Cited by 304 (18 self)
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The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
Faulttolerant quantum computation by anyons
, 2003
"... A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation ..."
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Cited by 229 (3 self)
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A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is faulttolerant by its physical nature.
Reliable quantum computers
 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
, 1998
"... The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mist ..."
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Cited by 165 (3 self)
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The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 106 qubits, with a probability of error per quantum gate of order 106, would be a formidable factoring engine. Even a smaller lessaccurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence
Quantum circuits with mixed states
 in Proc. 30th STOC
, 1998
"... Current formal models for quantum computation deal only with unitary gates operating on “pure quantum states”. In these models it is difficult or impossible to deal formally with several central issues: measurements in the middle of the computation; decoherence and noise, using probabilistic subrout ..."
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Cited by 142 (7 self)
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Current formal models for quantum computation deal only with unitary gates operating on “pure quantum states”. In these models it is difficult or impossible to deal formally with several central issues: measurements in the middle of the computation; decoherence and noise, using probabilistic subroutines, and more. It turns out, that the restriction to unitary gates and pure states is unnecessary. In this paper we generalize the formal model of quantum circuits to a model in which the state can be a general quantum state, namely a mixed state, or a “density matrix”, and the gates can be general quantum operations, not necessarily unitary. The new model is shown to be equivalent in computational power to the standard one, and the problems mentioned above essentially disappear. The main result in this paper is a solution for the subroutine problem. The general function that a quantum circuit outputs is a probabilistic function. However, the question of using probabilistic functions as subroutines was not previously dealt with, the reason being that in the language of pure states, this simply can not be done. We define a natural notion of using general subroutines, and show that using general subroutines does not strengthen the model. As an example of the advantages of analyzing quantum complexity using density matrices, we prove a simple lower bound on depth of circuits that compute probabilistic functions. Finally, we deal with the question of inaccurate quantum computation with mixed states. Using the so called “trace metric ” on density matrices, we show how to keep track of errors in the new model.
The Heisenberg representation of quantum computers, talk at
 International Conference on Group Theoretic Methods in Physics
, 1998
"... Since Shor’s discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers — the difficulty of describing them on classical computers — also makes it diff ..."
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Cited by 101 (2 self)
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Since Shor’s discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers — the difficulty of describing them on classical computers — also makes it difficult to describe and understand precisely what can be done with them. A formalism describing the evolution of operators rather than states has proven extremely fruitful in understanding an important class of quantum operations. States used in error correction and certain communication protocols can be described by their stabilizer, a group of tensor products of Pauli matrices. Even this simple group structure is sufficient to allow a rich range of quantum effects, although it falls short of the full power of quantum computation. 1
A personal view of averagecase complexity
 in 10th IEEE annual conference on structure in complexity theory, IEEE computer society press. Washington DC
, 1995
"... The structural theory of averagecase complexity, introduced by Levin, gives a formal setting for discussing the types of inputs for which a problem is dicult. This is vital to understanding both when a seemingly dicult (e.g. NPcomplete) problem is actually easy on almost all instances, and to d ..."
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Cited by 84 (1 self)
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The structural theory of averagecase complexity, introduced by Levin, gives a formal setting for discussing the types of inputs for which a problem is dicult. This is vital to understanding both when a seemingly dicult (e.g. NPcomplete) problem is actually easy on almost all instances, and to determining which problems might be suitable for applications requiring hard problems, such as cryptography. This paper attempts to summarize the state of knowledge in this area, including some \folklore " results that have not explicitly appeared in print. We also try to standardize and unify denitions. Finally, we indicate what we feel are interesting research directions. We hope that this paper will motivate more research in this area and provide an introduction to the area for people new to it.