Parallel Quantum Computation and Quantum Codes

Moore, Cristopher; Nilsson, Martin

doi:10.1137/s0097539799355053

Cited by 142 publications

(139 citation statements)

References 16 publications

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“…The problem for implementation will be creating an efficient MUX, especially on N T C. Figure 4 makes it clear that the total carry-select adder is only faster if the latency of MUX is substantially less than the latency of the full carry-ripple. It will be difficult for this to be more efficient that the single-CCNOT delay of the basic VBE carry-ripple adder on N T C. On AC, it is certainly easy to see how the MUX can use a fanout tree consisting of more ancillae and CNOT gates to distribute the carry in signal, as suggested by Moore [12], allowing all MUX Fredkin gates to be executed concurrently. A full fanout requires an extra m qubits in each adder.…”

Section: O( √ N) Carry-select Addermentioning

confidence: 99%

“…Circuit depth is explicitly considered in Cleve and Watrous' parallel implementation of the quantum Fourier transform [9], Gossett's quantum carry-save arithmetic [10], and Zalka's Schönhage-Strassen-based implementation [11]. Moore and Nilsson define the computational complexity class QNC to describe certain parallelizable circuits, and show which gates can be performed concurrently, proving that any circuit composed exclusively of Control-NOTs (CNOTs) can be parallelized to be of depth O(log n) using O(n 2 ) ancillae on an abstract machine [12].…”

Section: Introductionmentioning

confidence: 99%

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Fast quantum modular exponentiation

2005

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We present a detailed analysis of the impact on quantum modular exponentiation of architectural features and possible concurrent gate execution. Various arithmetic algorithms are evaluated for execution time, potential concurrency, and space tradeoffs. We find that to exponentiate an n-bit number, for storage space 100n (twenty times the minimum 5n), we can execute modular exponentiation two hundred to seven hundred times faster than optimized versions of the basic algorithms, depending on architecture, for n = 128. Addition on a neighboronly architecture is limited to O(n) time while non-neighbor architectures can reach O(log n), demonstrating that physical characteristics of a computing device have an important impact on both real-world running time and asymptotic behavior. Our results will help guide experimental implementations of quantum algorithms and devices.

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Section: O( √ N) Carry-select Addermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fast quantum modular exponentiation

2005

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“…Using an Euler decomposition of SU (2), the diagonal gate can be can be built using 3(d − 1) Givens rotations.…”

Section: Parallelism In State Synthesis and Unitary Transformatiomentioning

confidence: 99%

“…A third aspect of parallelism [2] involves reducing the logical depth of a circuit by judicious grouping of single-and two-particle gates that can be performed at the same time step, assuming connectivity of the particles. This is roughly analogous to classic circuit layouts and will not be considered here.…”

Section: Introductionmentioning

confidence: 99%

Parallelism for quantum computation with qudits

O’Leary

Brennen

Bullock³

2006

Phys. Rev. A

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Robust quantum computation with d-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high fidelity two-qudit gates. We first describe how to implement parallel single qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the atomic alkalies 87 Rb and 133 Cs. Second, we provide a protocol for implementing parallelized nonlocal two-qudit gates using the assistance of entangled qubit pairs. Because the entangled qubits can be prepared non-deterministically, this offers the possibility of high fidelity two-qudit gates.

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“…Also, 3D lattices or alternative 2D lattices such as hexagonal close packing may be advantageous, especially with regard to resource scheduling. For instance, some quantum algorithms can be parallelized to exploit the commutivity of certain operations [18] if pairs of memory nodes can be simultaneously connected. It is not clear what the optimal scheme is to create the necessary entanglement resources simultaneously to perform such nonlocal operations since the resource of bus qubits is limited.…”

mentioning

confidence: 99%

Quantum-computer architecture using nonlocal interactions

2003

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Several authors have described the basic requirements essential to build a scalable quantum computer. Because many physical implementation schemes for quantum computing rely on nearest neighbor interactions, there is a hidden quantum communication overhead to connect distant nodes of the computer. In this paper we propose a physical solution to this problem which, together with the key building blocks, provides a pathway to a scalable quantum architecture using nonlocal interactions. Our solution involves the concept of a quantum bus that acts as a refreshable entanglement resource to connect distant memory nodes providing an architectural concept for quantum computers analogous to the von Neumann architecture for classical computers. Most modern computers share the same basic architecture first proposed by von Neumann in 1945. Von Neumann organized a computer into four basic components: memory, an input/output system, an arithmetic logic unit, and a control unit. The four units were interconnected by a bus that provided for the flow of classical bits or information between the various components [1]. Basic elements sufficient to build a scalable quantum computer have been described by DiVincenzo [2] and Preskill [3]. The five DiVincenzo criteria [2] for building a quantum computer are: a scalable physical system with well characterized qubits, the ability to initialize the state of the qubits to a simple fiducial state, long relevant decoherence times, a universal set of quantum gates, and a qubit specific measurement capability. In addition, Preskill lists other elements necessary for fault tolerant computation in order to maintain a reasonable accuracy threshold. Two of these are maximal parallelism and gates that can act on any pair of qubits.Although Preskill [3] communicates the need to interact arbitrary pairs of qubits, he provides no solution for this in a typical quantum computer restricted to nearest neighbor interactions. DiVincenzo [2] mentions two additional criteria essential for quantum communications namely: the ability to interconvert stationary and flying qubits, and the ability to faithfully transmit flying qubits between specified locations. Clearly, if such capabilities were engineered into the architecture, the above requirements of qubit interconnectivity and parallelism could simultaneously be satisfied. In this paper we show an alternative approach based on the concept of a quantum bus that consists of refreshable qubits that act as a resource for entanglement. This concept bears similarity to the classical bus, key to the von Neumann architecture.For concreteness we consider a lattice model of a quantum computer (e.g. a neutral atom optical lattice, quantum dot arrays, or 31 P embedded Si [4]) where qubits are fixed in position and interactions are with nearest neighbors (Fig. 1). One obvious way to connect distant qubits is to swap the states through intermediary qubits until the states are adjacent to each other, perform the requisite operations, and then swap back. This procedure s...

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Parallel Quantum Computation and Quantum Codes

Cited by 142 publications

References 16 publications

Fast quantum modular exponentiation

Fast quantum modular exponentiation

Parallelism for quantum computation with qudits

Quantum-computer architecture using nonlocal interactions

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