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PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 1278 (4 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
Elementary Gates for Quantum Computation
, 1995
"... We show that a set of gates that consists of all onebit quantum gates (U(2)) and the twobit exclusiveor gate (that maps Boolean values (x, y)to(x, x⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n)) can be expressed as compositions of these gates. We in ..."
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Cited by 279 (11 self)
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We show that a set of gates that consists of all onebit quantum gates (U(2)) and the twobit exclusiveor gate (that maps Boolean values (x, y)to(x, x⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n)) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized DeutschToffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two and threebit quantum gates, the asymptotic number required for nbit DeutschToffoli gates, and make some observations about the number required for arbitrary nbit unitary operations.
Universal quantum gates
 in Mathematics of Quantum Computation, Chapman & Hall/CRC Press, Boca
, 2002
"... Abstract. In this paper we study universality for quantum gates acting on qudits. Qudits are states in a Hilbert space of dimension d where d can be any integer ≥ 2. We determine which 2qudit gates V have the properties (i) the collection of all 1qudit gates together with V produces all nqudit ga ..."
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Cited by 30 (0 self)
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Abstract. In this paper we study universality for quantum gates acting on qudits. Qudits are states in a Hilbert space of dimension d where d can be any integer ≥ 2. We determine which 2qudit gates V have the properties (i) the collection of all 1qudit gates together with V produces all nqudit gates up to arbitrary precision, or (ii) the collection of all 1qudit gates together with V produces all nqudit gates exactly. We show that (i) and (ii) are equivalent conditions on V, and they hold if and only if V is not a primitive gate. Here we say V is primitive if it transforms any decomposable tensor into a decomposable tensor. We discuss some applications and also relations with work of other authors. 1. Statements of main results We determine which 2qudit gates V have the property that all 1qudit gates together with V form a universal collection, in either the approximate sense or the exact sense. Here d is an arbitrary integer ≥ 2. Our results are new for the case of qubits, i.e., d = 2 (which for many is the case of primary interest). We treat the case d> 2 as well because it is of independent interest and requires no additional work.
Cartan decomposition of SU(2 n ), constructive controllability of spin systems and universal quantum computing
 Chem. Physics
, 2001
"... In this paper we provide an explicit construction of any arbitrary unitary transformation on n qubits from one qubit and a single two qubit quantum gate. Building on the previous work demonstrating the universality of two qubit quantum gates, we present here an explicit implementation. The construct ..."
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Cited by 9 (0 self)
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In this paper we provide an explicit construction of any arbitrary unitary transformation on n qubits from one qubit and a single two qubit quantum gate. Building on the previous work demonstrating the universality of two qubit quantum gates, we present here an explicit implementation. The construction is based on the Cartan decomposition of the semisimple Lie group SU(2n) and uses the SU(2 n) SU(2 n−1)⊗SU(2 n−1)⊗U(1) geometric structure of the Riemannian Symmetric Space. The decomposition highlights the geometric aspects of the problem of building an arbitrary unitary transformations out of quantum gates and makes explicit the choice of pulse sequences for the implementation of arbitrary unitary transformation on n coupled 1 2 spins in NMR quantum computing. Finally we make observations on the optimality of the design procedure. 1
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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Cited by 9 (2 self)
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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.
Controllability of Timedependent Quantum Control Systems
, 2003
"... Abstract — The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinitedimension ..."
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Cited by 5 (1 self)
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Abstract — The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinitedimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finitedimensional submanifold of the unit sphere in Hilbert space – and thus on analytic controllability. A relatively straightforward strategy allows the extension of Liealgebraic conditions for strong analytic controllability derived earlier for the simpler, timeindependent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as timeindependent. Methods developed by Kunita can then be implemented to establish controllability conditions for the onedimensionreduced system defined by the original timedependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples. I.
A selfassembled nanoelectronic quantum computer based on the Rashba effect in quantum dots,” LANL quantph/9910032
 Physical Review B
"... Quantum computers promise vastly enhanced computational power and an uncanny ability to solve classically intractable problems. However, few proposals exist for robust, solid state implementation of such computers where the quantum gates are sufficiently miniaturized to have nanometerscale dimensio ..."
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Quantum computers promise vastly enhanced computational power and an uncanny ability to solve classically intractable problems. However, few proposals exist for robust, solid state implementation of such computers where the quantum gates are sufficiently miniaturized to have nanometerscale dimensions. Here I present a new approach whereby a complete computer with nanoscale gates might be selfassembled using chemical synthesis. Specifically, I demonstrate how to selfassemble the fundamental unit of this quantum computer – a 2qubit universal quantum gate – based on two exchange coupled multilayered quantum dots. Then I show how these gates can be wired using thiolated conjugated molecules as electrical connectors. Each quantum dot in this architecture consists of ferromagnetsemiconductorferromagnet layers. The ground state in the semiconductor layer is spin split because of the Rashba interaction and the spin splitting energy can be varied by an external electrostatic potential applied to the dot. A spin polarized electron is injected into each dot from one of the ferromagnetic layers and trapped by Coulomb blockade. Its spin orientation encodes a qubit. Arbitrary qubit rotations are effected by bringing the spin splitting energy in a target quantum dot in resonance with a global ac magnetic field by applying a potential pulse of appropriate amplitude and duration to the dot. The controlled dynamics of the universal 2qubit rotation operation can be realized by exploiting the exchange coupling with the nearest neighboring dot. The qubit (spin orientation) is read via the current induced between the ferromagnetic layers under an applied potential. The ferromagnetic layers act as “polarizers ” and “analyzers ” for spin injection and detection. A complete prescription for initialization of the computer and data input/output operations is presented. This paradigm draws together two great recent scientific advances: one in materials science (nanoscale self assembly) and the other in
Another Way to Perform the Quantum Fourier Transform in Linear Parallel Time, ftp://ftp.santafe.edu/pub/moore/qft.ps; and references therein
"... Abstract. We exhibit a quantum circuit that performs the Quantum Fourier Transform on n qubits in O(n) depth. Thus, a parallel quantum computer can carry out the QFT in linear time. Griffiths and Niu have already shown this, so this paper is little more than an exercise in quantum circuit design; bu ..."
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Abstract. We exhibit a quantum circuit that performs the Quantum Fourier Transform on n qubits in O(n) depth. Thus, a parallel quantum computer can carry out the QFT in linear time. Griffiths and Niu have already shown this, so this paper is little more than an exercise in quantum circuit design; but perhaps it illustrates a worthwhile idea. We also speculate as to whether the QFT might be in the class QNC 1 of problems solvable in logarithmic parallel time. Shor’s factoring algorithm [6] suggests that quantum computers can do things in polynomial time that classical computers cannot. However, since decoherence due to storage errors is a function of time, we should also ask to what extent we can parallelize quantum algorithms; if we can do many quantum operations at once, rather than serially, we can solve larger problems before our computer decoheres. Consider a quantum circuit operating on a set of qubits, containing onequbit gates (2 ×2 unitary matrices) and the twoqubit controllednot gate; these are universal for quantum computation [1, 4]. We can define the depth of this circuit as the number of layers, where each layer consists of gates operating on mutually disjoint sets of qubits; that is, each qubit interacts with at most one other qubit at a time. (In a model of quantum computation where one qubit can simultaneously interact with several others, we could allow gates operating on the same qubit in the same level, as long as these gates all mutually commute.) The heart of Shor’s algorithm is the Quantum Fourier Transform. If we represent ndigit numbers a 〉 with n qubits, the QFT maps a 〉 to 2 −n/2 2 n ∑−1 e 2πiab/2n b〉 b=0 In this paper, we exhibit a circuit with depth O(n) for performing the QFT. Griffiths and Niu have already done this, in fact in a more natural way [3]. However, perhaps the reader will enjoy a new construction using slightly different ideas. The standard quantum algorithm for the QFT takes n(n − 1)/2 gates [2, 6]. One way to construct it is to reshuffle the rows of the matrix by putting thedigits of the input in reverse order. Then for n = 3, for instance, we have