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Physical Review A

General method for realizing the conditional phase-shift gate and a simulation of Grover’s algorithm in an ion-trap system

Shingo Fujiwara, Shuichi Hasegawa

Published Jan 31, 2005• 25 citations
Publisher

Abstract

It is well known that, in order to build the universal quantum circuit, one only needs one-qubit rotation gate and two-qubit controlled-NOT gate and until now quantum networks have been built from these gates. However, the minimum components of quantum networks in real experiments are not these quantum gates, so we develop a general method for realizing the conditional phase-shift gate in multiqubit ion-trap quantum computation which has the scalability to $N$ qubits $(N\ensuremath{\geqslant}3)$. The duration of the laser manipulations for the proposed conditional phase-shift gate is almost the same as that for the controlled-NOT gate in ion-trap quantum computation. Moreover, we simulate Grover's algorithm taking into consideration the real laser fluctuations and analyze the effect of decoherence on the practical search.

DOI
10.1103/physreva.71.012337

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It is well known that, in order to build the universal quantum circuit, one only needs one-qubit rotation gate and two-qubit controlled-NOT gate and until now quantum networks have been built from these gates. However, the minimum components of quantum networks in real experiments are not these quantum gates, so we develop a general method for realizing the conditional phase-shift gate in multiqubit ion-trap quantum computation which has the scalability to NNN qubits (N\ensuremath⩾3)(N\ensuremath{\geqslant}3)(N\ensuremath⩾3). The duration of the laser manipulations for the proposed conditional phase-shift gate is almost the same as that for the controlled-NOT gate in ion-trap quantum computation. Moreover, we simulate Grover's algorithm taking into consideration the real laser fluctuations and analyze the effect of decoherence on the practical search.
We propose a scheme for realizing controlled multiqubit quantum phase gates via the adiabatic evolution for trapped ions. Leveraging on the adiabatic operation, the scheme is tolerant to ambient noise and insensitive to the randomness of moderate fluctuations in experimental parameters. In our scheme, the ions are illuminated by lasers tuned to the first lower vibrational sideband or quasiresonant transition, the vibrational mode is not utilized as a data bus, and dark state evolution is required. The required gate operation time does not increase with the increasing number of qubits. This leads to a more efficient implementation of multiqubit gates than that of a series of one-qubit and two-qubit operations for quantum circuits and quantum algorithms.
We propose a potentially practical scheme for realization of an n-qubit (n>2) conditional phase flip (CPF) gate and implementation of Grover's search algorithm in an ion-trap system. We demonstrate, both analytically and numerically, that our scheme could be achieved efficiently to find a marked state with high fidelity and high success probability. We also show the merits of the proposal that the increase of the ion number can improve the fidelity and success probability of the CPF gate. The required operations for a Grover search are very close to the capabilities of current ion-trap techniques.
The implementation of a quantum computer requires the realization of a large number of N-qubit unitary operations which represent the possible oracles or which are part of the quantum algorithm. Until now there have been no standard ways to uniformly generate whole classes of N-qubit gates. We develop a method to generate arbitrary controlled phase-shift operations with a single network of one-qubit and two-qubit operations. This kind of network can be adapted to various physical implementations of quantum computing and is suitable to realize the Deutsch-Jozsa algorithm as well as Grover's search algorithm.
We show how to construct a multiqubit control gate on a quantum register of an arbitrary size N. This gate performs a single-qubit operation on a specific qubit conditioned by the state of other $N\ensuremath{-}1$ qubits. We provide an algorithm how to build up an array of networks consisting of single-qubit rotations and multiqubit control-NOT (CNOT) gates for the synthesis of an arbitrary entangled quantum state of N qubits. We illustrate the algorithm on a system of cold trapped ions. This example illuminates the efficiency of the direct implementation of the multiqubit CNOT gate compared to its decomposition into a network of two-qubit CNOT gates.
The Grover quantum search algorithm is a hallmark application of a quantum computer with a well-known speedup over classical searches of an unsorted database. Here, we report results for a complete three-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance. Two methods of state marking are used for the oracles: a phase-flip method employed by other experimental demonstrations, and a Boolean method requiring an ancilla qubit that is directly equivalent to the state marking scheme required to perform a classical search. We also report the deterministic implementation of a Toffoli-4 gate, which is used along with Toffoli-3 gates to construct the algorithms; these gates have process fidelities of 70.5% and 89.6%, respectively.
Any residual coupling of a quantum computer to the environment results in computational errors. Encoding quantum information in a so-called decoherence-free subspace provides means to avoid these errors. Despite tremendous progress in employing this technique to extend memory storage times by orders of magnitude, computation within such subspaces has been scarce. Here, we demonstrate the realization of a universal set of quantum gates acting on decoherence-free ion qubits. We combine these gates to realize the first controlled-NOT gate towards a decoherence-free, scalable quantum computer.