Geometric quantum computation

Ekert, Artur; Ericsson, Marie; Hayden, Patrick; Inamori, Hitoshi; Jones, Jonathan A.; Oi, Daniel K. L.; Vedral, Vlatko

doi:10.1080/095003400750039500

Cited by 68 publications

(85 citation statements)

References 4 publications

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“…A similar approach was adopted by Jones et.al. to demonstrate the construction of controlled phase shift gates in a two-qubit system using adiabatic geometric phase [13,14]. Pines et.al.…”

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confidence: 99%

Use of non-adiabatic geometric phase for quantum computing by NMR

Das

Kumar

2005

Journal of Magnetic Resonance

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Geometric phases have stimulated researchers for its potential applications in many areas of science. One of them is fault-tolerant quantum computation. A preliminary requisite of quantum computation is the implementation of controlled logic gates by controlled dynamics of qubits. In controlled dynamics, one qubit undergoes coherent evolution and acquires appropriate phase, depending on the state of other qubits. If the evolution is geometric, then the phase acquired depend only on the geometry of the path executed, and is robust against certain types of errors. This phenomenon leads to an inherently fault-tolerant quantum computation. Here we suggest a technique of using non-adiabatic geometric phase for quantum computation, using selective excitation. In a two-qubit system, we selectively evolve a suitable subsystem where the control qubit is in state |1 , through a closed circuit. By this evolution, the target qubit gains a phase controlled by the state of the control qubit. Using these geometric phase gates we demonstrate implementation of Deutsch-Jozsa algorithm and Grover's search algorithm in a two-qubit system.

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“…A similar approach was adopted by Jones et.al. to demonstrate the construction of controlled phase shift gates in a two-qubit system using adiabatic geometric phase [13,14]. Pines et.al.…”

mentioning

confidence: 99%

Use of non-adiabatic geometric phase for quantum computing by NMR

Das

Kumar

2005

Journal of Magnetic Resonance

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“…If the former are worse, for example due to additional external source terms in the Hamiltonian, then an optimistic upper bound on entanglement results. Despite these constraints, the simulation approach could be very useful for new experimental QC investigations.To illustrate the approach, we apply it to geometric phase gates [9,10,11]. This provides an example of the approach; however, the specific case of geometric phase gates is of interest in its own right, since the technique has already been applied to NMR experiments [9], has been proposed for use with superconducting qubits [11] and, in principle, can be applied to other realisations of qubits with suitable source terms in their Hamiltonians.…”

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confidence: 99%

Decoherence of geometric phase gates

2002

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We consider the effects of certain forms of decoherence applied to both adiabatic and non-adiabatic geometric phase quantum gates. For a single qubit we illustrate path-dependent sensitivity to anisotropic noise and for two qubits we quantify the loss of entanglement as a function of decoherence.PACS numbers: 03.67. Lx, 03.65.Vf The discoveries of quantum algorithms for factorization [1] and searching [2], and techniques for quantum error correction [3,4] (and subsequent fault-tolerant methods) have generated considerable motivation for the realisation of quantum computing (QC) hardware. Significant progress has been made at the few qubit level; at present many possible alternative routes are under exploration [5]. The use of fundamental entities as qubits currently leads the way; however, there is growing longer term hope that fabricated condensed matter systems (maybe still using fundamental rather than fabricated qubits) may provide the vehicle for scalability in qubit number. Nevertheless, at present the decoherence problems in such systems loom very large indeed. Faulttolerant techniques cannot be brought into play unless the underlying decoherence rates are small in the first place.As arbitrary single-qubit gates and some entangling two-qubit gate are universal for quantum computing [6,7,8], experimental QC focuses on realising such gates. Whilst it would be unfair to deem single-qubit gates trivial, the entangling of qubits represents the first major hurdle-one cannot claim to have a serious QC candidate until this has been achieved. Nevertheless, with any new contender it is natural to investigate the simplest gates first, before progressing to those based on qubit coupling. Clearly a method for inferring the likely level of entanglement in a two-qubit gate from the results of single qubit experiments is a handy tool. This is the basis of the simulation approach we present here in the form of a specific example. It is certainly true that detailed calculations of decoherence effects (based on tracing over microscopic environments) can yield considerable understanding of the form of decoherence seen by a qubit, and can sometimes give estimates of decoherence rates; however, in any experimental realisation the ultimate calibration of decoherence is through measurement. Given this, we consider a simple simulation approach which, based on the observations of single qubit gates, enables estima- * Present address: Department of Materials, University of Oxford, OX1 3PH, UK.tion of the level of entanglement (if any!) that could be expected in a two-qubit gate with similar qubits. This relies on some advanced knowledge of the form of the dominant decohering mechanisms (although some of this may be inferred from various independent single-qubit experiments) and an assumption of environment effects in interacting two-qubit experiments being similar to those in the single-qubit experiments. If the former are worse, for example due to additional external source terms in the Hamiltonian, then an optimistic upper bou...

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“…The geometric phases that only depend on the path followed by the system during its evolution, have been investigated and tested in a variety of settings and have been generalized in several directions [3]. The geometric phases are attractive both from a theoretical perspective, and from the point of view of possible applications, among which geometric quantum computation [4,5,6,7] is one of the most importance.As realistic systems always interact with their environment, the study on the geometric phase in open systems become interesting. Garrison and Wright [8] were the first to touch on this issue by describing open system evolution in terms of a non-Hermitian Hamiltonian.…”

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confidence: 99%

Geometric phase in dephasing systems

Wang

2005

Phys. Rev. A

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Beyond the quantum Markov approximation, we calculate the geometric phase of a two-level system driven by a quantized magnetic field subject to phase dephasing. The phase reduces to the standard geometric phase in the weak coupling limit and it involves the phase information of the environment in general. In contrast with the geometric phase in dissipative systems, the geometric phase acquired by the system can be observed on a long time scale. We also show that with the system decohering to its pointer states, the geometric phase factor tends to a sum over the phase factors pertaining to the pointer states.PACS numbers: 03.65. Vf, 03.65.Yz Quantum mechanics tell us that physical states are equivalent up to a global phase, which in general does not contain useful information about the described system and thus can be ignored. This is not the case, however, for a system transported round a circuit by varying the parameters s = (s 1 , s 2 , ...) in its Hamiltonian H( s). As Berry showed [1], the phase can have a component of geometric origin called geometric phase with important observable consequences, such as the Aharonov-Bohm effect [2] and the spin-1 2 particle driven by a rotating magnetic field [1]. The geometric phases that only depend on the path followed by the system during its evolution, have been investigated and tested in a variety of settings and have been generalized in several directions [3]. The geometric phases are attractive both from a theoretical perspective, and from the point of view of possible applications, among which geometric quantum computation [4,5,6,7] is one of the most importance.As realistic systems always interact with their environment, the study on the geometric phase in open systems become interesting. Garrison and Wright [8] were the first to touch on this issue by describing open system evolution in terms of a non-Hermitian Hamiltonian. This is a pure state analysis, so it did not address the problem of geometric phases for mixed states. Toward the geometric phase for mixed states in open systems, the approaches used involve solving the master equation of the system [9,10,11,12,13], employing a quantum trajectory analysis [14,15] or Krauss operators [16], and the perturbative expansions [17,18]. Some interesting results were achieved, briefly summarized as follows: nonhermitian Hamiltonian lead to a modification of Berry's phase [8,17], stochastically evolving magnetic fields produce both energy shift and broadening [18], phenomenological weakly dissipative Liouvillians alter Berry's phase by introducing an imaginary correction [11] or lead to damping and mixing of the density matrix elements [12]. However, almost all these studies are performed for dissipative systems, and thus the representations are applicable for systems whose energy is not conserved. For open systems with conserved energy (dephasing systems), the problem beyond the Markov approximation remains untouched to our best knowledge. Because the systemenvironment interaction H I and the free system Hamilton...

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Geometric quantum computation

Cited by 68 publications

References 4 publications

Use of non-adiabatic geometric phase for quantum computing by NMR

Use of non-adiabatic geometric phase for quantum computing by NMR

Decoherence of geometric phase gates

Geometric phase in dephasing systems

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