Landau-Zener Transitions in the Presence of Spin Environment

Wan, Andy T. S.; Amin, M. H. S.; Wang, Shannon X.

doi:10.1142/s0219749909005353

Cited by 6 publications

(7 citation statements)

References 18 publications

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“…The groups of avoided-level crossing are formed not only between the states |↑ n and |↓ n + 1 but also between |↑ n and |↓ n − 1 (when n ≥ 2), with level splittings ∆ √ n + 1 and ∆ √ n, respectively. Note that the two groups of avoided crossings are approximately independent whenever the time between the successive avoided level crossings, t cross = 2ω/v, exceeds the duration of an individual LZ transition, τ LZ ∼ max {1/ √ v, ∆/v} [47,48]. For our multilevel LZ problem, the couplings …”

Section: A Ys Coherent State Casementioning

confidence: 98%

“…If all the avoided crossings are well separated, we can approximately treat the transitions as being independent and compute the transition probabilities as joint proba- bilities [28,48,49]. For the Hamiltonian (2) without the RWA, the final probability to find the TLS at |↑ from the initial state |↑ n is [49] P ↑,n→↑ = P ↑,n−1 P ↑,n + (1 − P ↑,n−1 ) (1 − P ↑,n−2 ) , (19) where P ↑,n = exp[−π∆ 2 (n + 1)/2v] = P n+1 ↑,0 , and the expanded form of the equation above still holds for the n = 0 and 1 cases.…”

Section: A Ys Coherent State Casementioning

confidence: 99%

“…bilities [28,48,49]. For the Hamiltonian (2) without the RWA, the final probability to find the TLS at |↑ from the initial state |↑ n is [49] P ↑,n→↑ = P ↑,n−1 P ↑,n + (1 − P ↑,n−1 ) (1 − P ↑,n−2 ) , (19) where P ↑,n = exp[−π∆ 2 (n + 1)/2v] = P n+1 ↑,0 , and the expanded form of the equation above still holds for the n = 0 and 1 cases.…”

Section: A Ys Coherent State Casementioning

confidence: 99%

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Photon-assisted Landau-Zener transition: Role of coherent superposition states

Sun

Wang

et al. 2012

Phys. Rev. A

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We investigate a Landau-Zener (LZ) transition process modeled by a quantum two-level system (TLS) coupled to a photon mode when the bias energy is varied linearly in time. The initial state of the photon field is assumed to be a superposition of coherent states, leading to a more intricate LZ transition. Applying the rotating-wave approximation (RWA), analytical results are obtained revealing the enhancement of the LZ probability by increasing the average photon number. We also consider the creation of entanglement and the change of photon statistics during the LZ process. Without the RWA, we find some qualitative differences of the LZ dynamics from the RWA results, e.g., the average photon number no longer monotonically enhances the LZ probability. The ramifications and implications of these results are explored.

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Section: A Ys Coherent State Casementioning

confidence: 98%

Section: A Ys Coherent State Casementioning

confidence: 99%

Section: A Ys Coherent State Casementioning

confidence: 99%

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Photon-assisted Landau-Zener transition: Role of coherent superposition states

Sun

Wang

et al. 2012

Phys. Rev. A

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“…In the appendix of Ref. [9], the effective Hamiltonian (15) is systematically derived for the case of adiabatic Grover search problem [16]. For that problem, one finds Q…”

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confidence: 99%

Role of single-qubit decoherence time in adiabatic quantum computation

2009

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We have studied numerically the evolution of an adiabatic quantum computer in the presence of a Markovian ohmic environment by considering Ising spin glass systems with up to 20 qubits independently coupled to this environment via two conjugate degrees of freedom. The required computation time is demonstrated to be of the same order as that for an isolated system and is not limited by the single-qubit decoherence time T * 2 , even when the minimum gap is much smaller than the temperature and decoherence-induced level broadening. For small minimum gap, the system can be described by an effective two-state model coupled only longitudinally to environment.Adiabatic quantum computation [1] (AQC) is an attractive model of quantum computation (QC). It eliminates the need for precise timing of the qubit transformations required in the gate-model computation scheme, and also is expected to possess some degree of fault tolerance afforded by the energy gap separating the ground from excited states of the qubit Hamiltonian. AQC approach is particularly appealing in the context of superconducting qubits which in principle have the required flexibility for implementation of complicated interactions. In the AQC, a system starts from a readily accessible ground state of some initial Hamiltonian H i and slowly evolves into the ground state of the final Hamiltonian H f which encodes solution to the problem of interest:where s(t) ∈ [0, 1] is a monotonic function of time t. Here, we only consider a linear time sweep s(t) = t/t f , where t f is the total evolution time. Transitions out of the ground state can be caused by the Landau-Zener processes [2] at the anticrossing (s=s * ), where the gap g between the ground state |0 and first excited state |1 goes through a minimum: g m ≡ g(s * ). The probability of being in the ground state at the end of the adiabatic evolution is approximately (h = k B = 1)To ensure large P 0f , one needs t f > ∼ t a . The computation time is hence determined by t a and thus by g m .In the gate-model QC, there is no direct correspondence between the wavefunction and the instantaneous system Hamiltonian. The Hamiltonian is only applied at the time of gate operations and usually involves only a few qubits. The wavefunction, therefore, is strongly affected by the environment and is irreversibly altered after the decoherence time, which is typically smaller than the single-qubit dephasing time T * 2 . This means that T * 2 imposes an upper limit on the total computation time, unless some quantum error correction scheme (which requires significant resources) is utilized. This is not true for AQC, as the wavefunction is always very close to the instantaneous ground state of the system Hamiltonian and is consequently more stable against the decoherence. Qualitatively, one expects decoherence to drive the system's reduced density matrix towards being diagonal in the energy basis, which is not harmful for AQC but is detrimental for the gate-model QC. Such robustness has been demonstrated in previous studies [3,4,...

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“…First, the above consideration ignores the effect of the environment, especially the fact that the lowest feasible temperature at which such a processor can be operated is larger than the median minimum energy gaps of problems with 48 or more variables. Although recent calculations suggest that a weak coupling to the environment does not significantly affect the time required to reach the final ground state with a measurable probability, even if the minimum gap is below the temperature [17,[31][32][33][34][35][36][37], a fair comparison should consider the performance of an open and not closed system. Unfortunately, such open system simulations are not feasible beyond ∼ 20 qubits [17].…”

Section: Discussionmentioning

confidence: 99%

Investigating the performance of an adiabatic quantum optimization processor

Karimi

Dickson

Hamze

et al. 2011

Quantum Inf Process

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Adiabatic quantum optimization offers a new method for solving hard optimization problems. In this paper we calculate median adiabatic times (in seconds) determined by the minimum gap during the adiabatic quantum optimization for an NP-hard Ising spin glass instance class with up to 128 binary variables. Using parameters obtained from a realistic superconducting adiabatic quantum processor, we extract the minimum gap and matrix elements using high performance Quantum Monte Carlo simulations on a large-scale Internet-based computing platform. We 123 78 K. Karimi et al. compare the median adiabatic times with the median running times of two classical solvers and find that, for the considered problem sizes, the adiabatic times for the simulated processor architecture are about 4 and 6 orders of magnitude shorter than the two classical solvers' times. This shows that if the adiabatic time scale were to determine the computation time, adiabatic quantum optimization would be significantly superior to those classical solvers for median spin glass problems of at least up to 128 qubits. We also discuss important additional constraints that affect the performance of a realistic system.

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Landau-Zener Transitions in the Presence of Spin Environment

Cited by 6 publications

References 18 publications

Photon-assisted Landau-Zener transition: Role of coherent superposition states

Photon-assisted Landau-Zener transition: Role of coherent superposition states

Role of single-qubit decoherence time in adiabatic quantum computation

Investigating the performance of an adiabatic quantum optimization processor

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