Abstract
Holonomic quantum computation (HQC) may not show its full potential in quantum speedup due to the prerequisite of a long coherent runtime imposed by the adiabatic condition. Here we show that the conventional HQC can be dramatically accelerated by using external control fields, of which the effectiveness is exclusively determined by the integral of the control fields in the time domain. This control scheme can be realized with net zero energy cost and it is faulttolerant against fluctuation and noise, significantly relaxing the experimental constraints. We demonstrate how to realize the scheme via decoherencefree subspaces. In this way we unify quantum robustness merits of this faulttolerant control scheme, the conventional HQC and decoherencefree subspace, and propose an expedited holonomic quantum computation protocol.
Introduction
As building blocks for quantum computers, the implementation of quantum gates has received considerable research efforts over the recent years^{1}. It has been reported experimentally that numbers of pulsecontrolled microscopic systems, such as solidstate spins^{2} and trapped ions^{3}, can be hosts for implementation of quantum gates. While enormous theoretical strategies for conventional quantum gate implementation have been proposed, there is a revived interest in using geometric phases to perform circuitbased quantum computation, termed as holonomic quantum computation (HQC)^{4}, which is enabled by the adiabatic quantum theorem. The theorem asserts that at any instant a quantum system remains nearby in its instantaneous eigenstate of a slowvarying Hamiltonian, specifically for a cyclic adiabatic process, a geometric phase (the Berry’s phase), is acquired over the course of the cycle^{5}. The geometric phase is exclusively determined by the trajectory of the system in its parameter space and robust against local fluctuation^{6,7}. Consequently, a geometric strategy for implementation of quantum gates permits faulttolerant and robust quantum information processing. Besides inherent resilience in nonAbelian geometric phases^{8}, HQC has an appealing advantage^{9,10,11} in utilizing the stateofart experimental setups due to its close relationship to the circuit model^{12,13,14}. A recent experiment has implemented a universal set of geometric quantum logic gates with diamond nitrogenvacancy centers^{15}, and evidently it will greatly promote research endeavour along this line.
The heart of HQC is the experimental implementation of the geometric phase acquired in a cyclic adiabatic passage. Despite its advantages, the geometric protocol itself is challenged with a dilemma. On one hand, any HQC algorithm requires a long characteristic runtime in order to satisfy the adiabatic condition^{16}. On the other hand, decoherence or leakage accumulated in this long runtime gives rise to errors in the HQC processing and may eventually destroy the quantumness of the system. To get rid of the dilemma, researchers have proposed several different protocols. Over a decade ago, Wu, Zanardi and Lidar^{17} initiated a scheme by embedding HQC into a decoherencefree subspace (DFS). This combined HQCDFS scheme utilizes the virtues of both the faulttolerance of HQC and the robustness of DFS against collective dephasing noise based on the symmetry structure of the interaction between the system and its environment. However, the residual individual noise remains and ruins the quantum adiabatic passages during the long runtime. Later on the HQCDFS scheme was extended by considering the collective dephasing of two neighboring physical qubits^{18}. Whereas it is more feasible experimentally, this scheme has a more stringent requirement for the runtime. Recently a nonadiabatic HQCDFS scheme was suggested where the characteristic timescale is reduced by increasing the characteristic energy, at the cost of a harsh restriction for the runtime equal to the period of the system^{19}. However, the fault tolerance from adiabaticity therefore becomes obscure.
In this paper, we propose a novel and composite strategy to tackle the long runtime issue in the HQC protocols via accelerating the adiabatic passage in DFS. We explain the mechanism and show specifically that the characteristic timescale of the adiabatic process can be vastly reduced by means of external field control^{20}. Interestingly, it is found that the particular design or shape of a control function, such as regular, random, chaotic and even noisy pulse sequences, is not as decisive as it seems to be, but only the integral of the control function in the time domain plays the crucial role in speeding up the adiabatic passage, which greatly relaxes constraints on experimental implementation of these control functions. Remarkably, we further discover that our Hamiltonians in the adiabatic representation are periodical functionals of the integral of the control functions, resulting in a net zeroenergycost control scheme – a new mechanism that accelerates adiabatic passages with the same effectiveness. These lead to a new type of faulttolerance against control fluctuations.
Results
Decoherencefree subspace for qubit gates
Decoherencefree subspace is based on the symmetry structure of the systemenvironment interaction^{21,22,23,24,25}. Here we briefly recall the method to realize a universal set of quantum gates acting on the DFS as firstly proposed in ref. 17. To implement a onequbit quantum gate in DFS, we consider a four physical qubit system with the Hamiltonian , where , are the XY interactions and DzialoshinskiMoriya terms, is Pauli X(Y) matrix acting on the ith physical qubit and m, l = 1, 2, 3, 4. This Hamiltonian commutes with the operator , where is a Z Pauli matrix acting on ith physical qubit. By setting , where φ(t) is specifically designed for HQC, , and all other , the Hamiltonian becomes
The bases for DFS have been identified as eigenvectors of Z^{17}, as spanned by {0〉, 1〉, 2〉, 3〉}, where 0〉 = 0001〉 and 1〉 = 0010〉 constitute the two orthonormal states for a logical qubit and 2〉 = 1000〉 and 3〉 = 0100〉 serve as ancilla. This DFS scheme is robust against collective dephasing described by Z ⊗ B, where B is an arbitrary Hermitian bath operator. It is straightforwardly proven that in the DFS, the Hamiltonian (1) can be rewritten as
where θ(t) = tan^{−1}(J_{13}/J_{12}).
Holonomic quantum computation in DFS
Consider a quantum system whose dynamics is governed by a timedependent Hamiltonian H(t) with instantaneous eigenvectors E_{n}(t)〉 and eigenvalues E_{n}(t). The wave function ψ(t)〉 satisfies the Schrödinger equation and can be formally written as , where is the dynamical phase. If the Hamiltonian varies adiabatically and there is a nonvanishing gap between the interested eigenvalues, the system will remain in the corresponding instantaneous eigenstate. Consequently, a Berry’s phase is given when the system passes along a closed loop in the Hamiltonian parameter space, which is pathindependent. Without loss of generality, one can consider a case where the system is initially at the nth ground state E_{n}〉. It follows that in the adiabatic regime , where γ_{n}(t) is the Berry’s phase given by . Here we emphasize that for dark states with eigenenergy E_{n}(t) = 0, its dynamical phase vanishes and the remaining overall phase is a geometric phase.
Equipped with Eq. (2), we are ready to construct our expeditedHQCDFS scheme. To build up a onequbit gate in DFS, we consider a cyclic Hamiltonian with period of T. We first consider a single qubit phase gate. The Hamiltonian H_{1}(t) is formally given by Eq. (2) regarding , , where a is a dimensionless undetermined coefficient. The two dark states in the DFS for Hamiltonian H_{1}(t) read as D_{0}(t)〉 = 0〉 and , respectively.
In the adiabatic regime, under the unitary evolution , where is timeordering operator, the dark states D_{0}〉 and D_{1}〉 become
respectively, where γ_{j}(T) is the Berry’s phase for D_{j}〉, j = 0, 1. In this manner we achieve a onequbit phase gate by . Note that D_{j}(T)〉 = D_{j}(0)〉. The gate can be expressed by a diagonal matrix as . The two Berry’s phases for dark states are γ_{0}(T) = 0 and
where J_{0}(x) is a zero order Bessel function of the first kind, respectively.
This technique is also applicable in realization of a single σ_{x} qubit gate. To build this gate, we implement the Hamiltonian in the same DFS yet spanned by {+〉, −〉, 2〉, 3〉}, where . It is written as
In this case, the new dark states are D_{0}(t)〉 = +〉 and , respectively. The transformations of dark states under time evolution are still described by Eq. (3), and the qubit gate reads,
which becomes the σ_{x}gate when γ_{1}(T) = π. Now we turn to the twoqubit controlledphase (CPhase) gate in DFS. Since each logical qubit consists of four physical qubits, eight physical qubits are involved in implementing a two logicalqubit gate. Let us suppose that one can implement the Hamiltonian
The four dark states of the Hamiltonian employed in implementing CPhase gate are given by D_{0}(t)〉 = 0, 0〉, D_{1}(t)〉 = 0, 1〉, D_{2}(t)〉 = 1, 0〉, , respectively.
Over a period T, the Hamiltonian (7) drives these states into D_{0}(0)〉 → D_{0}(T)〉, D_{1}(0)〉 → D_{1}(T)〉, D_{2}(0)〉 → D_{2}(T)〉 and , so that the twoqubit gate is , where γ_{3}(T) = γ_{1}(T) in Eq. (4). Tuning the free parameter a, one can get an arbitrary phase gate at will, for example, γ_{3}(T) = π requires J_{0}(2a) = 0 at the first root a = 1.2024.
Control scheme
We now come to the case where the Hamiltonian H(t) is not in the adiabatic regime. Our scheme is to implement a control c(t) upon the strength of the Hamiltonian such that^{20,26}
We first show that as long as the control is sufficiently fast and strong (which will be specified later). the system evolution will behave in the same way as that in the adiabatic regime, specifically the wave function ψ(t)〉 becomes proportional to an instantaneous eigenstate of H(t). It is interesting to note that this control scheme hardly depends on the details of c(t) but its integral in the time domain, and is a new type of faulttolerance against control fluctuations. Consequently, the evolution of the corresponding dark states are shown to be a qualified workstation for HQC and this induced adiabaticity will be utilized to realized the expedited HQC in virtue of a fast modulation over Hamiltonian. We emphasize that the results given in Eq. (4) are invariant under the transformation (8), which is one of key points of our proposal.
To determine the effectiveness of our control scheme, we now introduce a quality factor
where δγ_{1} is the difference between the ideal phase (4) and the phase acquired during a finite runtime T. Note, δγ_{1} ≤ π because of the periodicity of the phase factor. Accordingly, due to our choice of quality factor (9) we have 0 ≤ f ≤ 1 where f = 1 if and only if the process is perfectly adiabatic and retain the Berry phase predicted by (4). Figure 1 shows f as a function of evolution time T (blue curve) in the absence of control (c(t) = 0), and as a function of average noise kick’s strength 〈c(t)〉 for T (red dashed curve) that is not in the adiabatic domain.
Discussion: Expedited HQC with net zero energy cost
On closely looking into its pattern, we find that the Hamiltonian (2) in the adiabatic representation (see Methods) is eventually a functional of the exponent e^{iC(t)}, i. e., . Because of the periodicity of e^{iC(t)}, our control scheme allows for an interesting case when 〈c′(t)〉 = 0, where c′(t) has alternating positive and negative values such that the net energy cost is zero. We first illustrate that the abovediscussed positive control c(t) (with ) can be exactly equivalent to zeroenergycost control c′(t), when c(t) = π∑_{i}δ(t − τ_{i}) with the integral C(t), and c′(t) = π∑_{i}(−1)^{i}δ (t − τ_{i}) with C′(t). It is easy to show that due to the periodicity of e^{iC(t)}, and 〈c(t)〉 = 2π/Δτ_{i} (Δτ_{i} = τ_{i+1} − τ_{i}) but 〈c′(t)〉 = 0 for each two consecutive pulses. The random intervals Δτ_{i} are much shorter than T in reality, and ideally the net energy cost of the c′(t) control sequence can be considered as zero when Δτ_{i} approaches zero. We can also analyze the equivalence for the rectangular pulses sequences. Based on the first order of Magnus^{27} expansion of U(δt) we can justify (see Methods) that if the single pulse strength , the offdiagonal terms in evolution U(δt) become zero when
In Fig. 2 we show the numerical simulation of the quality factor f for fixed T = 10 with net zeroenergycost control as a function of control pulse length Δt. We mark with triangles and squares when Eq. (10) is satisfied. The green solid curve in Fig. 2 shows the zeroenergycost noise control which is more robust against the control “kick” length Δt, while noise positive control has prominent oscillatory dependence on Δt which requires a more accurate choice of Δt (and/or J) according to Eq. (10). In Fig. 3 we show in detail how the quality factor f depends on JΔt/2π for different J values. Solid (dotted) lines correspond to the noiseless (noise) control. From Fig. 3 we can see that one can achieve f > 1–10^{−4} which is necessary for practical quantum computation by increasing strength J for both noise and noiseless control (note, the increase of J doesn’t change the average 〈c(t)〉 = 0 in netzeroenergy control). This circumstance follows from (10) which is derived based on the assumption , meaning that the increase of J automatically implies the decrease of Δt when n = Const.
Both HQC and DFS have been experimentally demonstrated in different physical systems. For instance, HQC was realized in the nuclear magnetic resonance (NMR)^{28}, trapped ions^{29} and superconducting qubit experiments^{30}. DFS has been experimentally demonstrated in NMR^{31}, trapped ions^{32} and photonic systems^{33}. Generalization of the DFS idea to noiseless subsystems^{34} was experimentally tested in ref. 35. The technique to implement the strength as in our control scheme has been developed and used, for example, in superconducting qubit experiments in ref. 30 by applying microwave pulses with timedependent envelope.
Methods
To understand the mechanism of our expedited HQC scheme we expand the wave function in terms of eigenstates E_{n}(t)〉 of the Hamiltonians. The matrix elements of the Hamiltonians in the adiabatic representation reads, ^{20}. For example, the Hamiltonian (2) is
where . It shows clearly that the Hamiltonian is a functional of the integral C(t) (or the average of c(t) in the time domain) i.e., , meaning that controlled dynamics does not depend on the details of c(t) but exclusively depends on the integral C(t). Such exclusive dependence also holds for our Hamiltonians (2), (5) and (7), and is a unique feature of our chosen Hamiltonians whose energy differences E_{nm} = E_{m} − E_{n} are timeindependent constants. These Hamiltonians, as shown in its adiabatic representation, are incidentally equivalent to the Leakage Elimination Operators^{36}. Hence, the control is fault tolerant in the sense that the fluctuation or noise of c(t) hardly contributes to C(t). More specially, by considering the propagator from t = 0 to t = δt, where and , we can write the propagator as, . The existence of the fast oscillating factor e^{iC(t)} renders all the offdiagonal elements of the propagator vanish and then leaves a Berry’s phase to the amplitudes of D_{1}〉 and two bright eigenstates. Noticeably this factor pushes the evolution of system into the adiabatic regime by decoupling all the four eigenstates. It clearly illustrates the advantage of our control scheme: one needs not to care about the exact control function because only the integral C(t) contributes to adiabaticity.
Expression (10) could be easily derived by the following consideration. We again consider short time evolution U(δt), and now we set , where Δt is control pulse length. Adiabaticity means that offdiagonal elements of the matrix are zero. Each of these offdiagonal elements could be written as , here we assume that and f(t) is a smooth function: f(t) ≈ f(t + Δt), sign ± corresponds to the positive and negative control pulses. We can then conclude that offdiagonal elements of U(Δt) becomes close to zero when JΔt = 2πn (n = 1, 2, 3, …), i.e. we have Δt equals to integer numbers of periods 2π/J.
Conclusion
To cope with the long runtime issue in implementing adiabatic passages, we have introduced an expeditedHQCDFS control scheme to accelerate the conventional HQC. We show explicitly that the effectiveness of our control scheme exclusively depends on the integral of the external control functions in the time domain. Therefore the scheme is robust against stochastic errors in control. More importantly, we further find that the Hamiltonian in the adiabatic representation is a periodical functional of the integral of the control. The periodicity motivates us to design a net zero energy cost strategy for speedup which is also robust against control imperfections. These novel results are confirmed by numerical results. This observation greatly reduces the experimental constraints in generating preciselyshaped pulses and allows us to use even random pulse sequences. By combining the features of this scheme with a scalable DFS, our expedited HQC protocol brings together the advantages of allgeometrical HQC, decoherencefree subspace, zeroenergycost control, and our fault tolerant scheme, a typical scalable, fast and faulttolerant architecture. We therefore expect that this perfect theoretical protocol becomes an experimental practice.
Additional Information
How to cite this article: Pyshkin, P. V. et al. Expedited Holonomic Quantum Computation via Net ZeroEnergyCost Control in DecoherenceFree Subspace. Sci. Rep. 6, 37781; doi: 10.1038/srep37781 (2016).
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Change history
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Acknowledgements
We acknowledge grant support from the Basque Government (grant IT47210), the Spanish MICINN (No. FIS201236673C0303), the NBRPC No. 2014CB921401, the NSAF No. U1330201, the NSFC Nos. 11575071 and 91421102, and Science and Technology Development Program of Jilin Province of China (20150519021JH).
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Affiliations
Beijing Computational Science Research Center, Beijing 100084, China
 P. V. Pyshkin
 , DaWei Luo
 , Jun Jing
 & J. Q. You
Department of Theoretical Physics and History of Science, The Basque Country University (EHU/UPV), PO Box 644, 48080 Bilbao, Spain
 P. V. Pyshkin
 , DaWei Luo
 & LianAo Wu
Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain
 P. V. Pyshkin
 , DaWei Luo
 & LianAo Wu
Institute of Atomic and Molecular Physics and Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin University, Changchun 130012, Jilin, China
 Jun Jing
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Contributions
P.V.P. contributed to numerical and physical analysis and prepared the first version of the manuscript and L.A.W. to the conception and design of this work. P.V.P., D.L., J.J., J.Q.Y. and L.W. discussed the results and implications at all stages, and wrote the manuscript.
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The authors declare no competing financial interests.
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Correspondence to LianAo Wu.
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1.
Experimental realization of singleshot nonadiabatic holonomic gates in nuclear spins
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