Andrew M. Childs, Edward Farhi, John Preskill
We study the fault tolerance of quantum computation by adiabatic evolution, a quantum algorithm for solving various combinatorial search problems. We describe an inherent robustness of adiabatic computation against two kinds of errors, unitary control errors and decoherence, and we study this robustness using numerical simulations of the algorithm.
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Adiabatic quantum computation for performing quantum computations such as Shor's algorithm is protected against thermal errors by an energy gap of size $O(1/n)$, where $n$ is the length of the computation to be performed.
We develop new protocols for high-fidelity single qubit gates that exploit and extend theoretical ideas for accelerated adiabatic evolution. Our protocols are compatible with qubit architectures with highly isolated logical states, where traditional approaches are problematic; a prime example are superconducting fluxonium qubits. By using an accelerated adiabatic protocol we can enforce the desired adiabatic evolution while having gate times that are comparable to the inverse adiabatic energy gap (a scale that is ultimately set by the amount of power used in the control pulses). By modelling t…
The robustness of the local adiabatic quantum search to decoherence in the instantaneous eigenbasis of the search Hamiltonian is examined. We demonstrate that the asymptotic time-complexity of the ideal closed case is preserved, as long as the Hamiltonian dynamics is present. In the special case of pure decoherence where the environment monitors the search Hamiltonian, it is shown that the local adiabatic quantum search performs as the classical search.
Shortcuts to adiabaticity (STA) are fast methods to realize the same final state evolution of quantum adiabatic process. We develop a general theory to construct a new kind of STA by solely sampling the points of the adiabatic path of the original adiabatic Hamiltonian. In contrast to previous methods, our approach does not require any additional control resources, has inherent robustness, and does not use any complicated unitary transformation. Based on path sampling, the resulting STA protocols are simple and can avoid points of adiabatic path that are challenging to implement. As applicatio…
The robustness to different sources of error of the scheme for non-adiabatic holonomic gates proposed in [New J. Phys. {\bf 14}, 103035 (2012)] is investigated. Open system effects as well as errors in the driving fields are considered. It is found that the gates can be made error resilient by using sufficiently short pulses. The principal limit of how short the pulses can be made is given by the breakdown of the quasi-monochromatic approximation. A comparison with the resilience of adiabatic gates is carried out.
Geometric phases are an interesting resource for quantum computation, also in view of their robustness against decoherence effects. We study here the effects of the environment on a class of one-qubit holonomic gates that have been recently shown to be characterized by "optimal" working times. We numerically analyze the behavior of these optimal points and focus on their robustness against noise.
Shortcuts to adiabaticity provide a general approach to mimic adiabatic quantum processes via arbitrarily fast evolutions in Hilbert space. For these counter-diabatic evolutions, higher speed comes at higher energy cost. Here, the counter-diabatic theory is employed as a minimal energy demanding scheme for speeding up adiabatic tasks. As a by-product, we show that this approach can be used to obtain infinite classes of transitionless models, including time-independent Hamiltonians under certain conditions over the eigenstates of the original Hamiltonian. We apply these results to investigate s…
We propose a method to design pulses in a resonant three-level system to enhance the robustness of non-adiabatic geometric gate operations. By optimizing the shape of the pulse envelope, we show that the gate operations are more robust against frequency detuning than they are with Gaussian and square pulses. Our method provides a way to design pulses that can be employed in a system where robustness against frequency variations or inhomogeneous broadening is required,and may be extended to ensure robustness against other physical imperfections such as intensity fluctuations and random noises.
Adiabatic operations are powerful tools for robust quantum control in numerous fields of physics, chemistry and quantum information science. The inherent robustness due to adiabaticity can, however, be impaired in applications requiring short evolution times. We present a single versatile gradient-based optimization protocol that combines adiabatic control with effective Hamiltonian engineering in order to design adiabatic operations tailored to the specific imperfections and resources of an experimental setup. The practicality of the protocol is demonstrated by engineering a fast, 2.3 Rabi cy…
Speeding up adiabatic holonomic quantum gates via $π$-pulse modulation
提出基于$π$脉冲调制的快速绝热完整子量子门方案，沿参数空间测地线引导哈密顿量并离散施加相位调制$π$脉冲，在保持绝热鲁棒性的同时实现通用单、双量子比特门。
We discuss applications of shortcuts to adiabaticity (STA) to adiabatic quantum computation. After reviewing the fundamental properties and the present status of STA from the author's personal point of view, we apply the method to the adiabatic algorithm of the Grover's problem. We discuss two possible implementations of STA for the adiabatic quantum computations: the method of quantum adiabatic brachistochrone and the Lewis-Riesenfeld invariant-based inverse engineering.
Adiabatic pulses are used extensively to enable robust control of quantum operations. We introduce a new approach to adiabatic control that uses the superadiabatic quality or $Q$-factor as a performance metric to design robust, high fidelity pulses. This approach permits the systematic design of quantum control schemes to maximize the adiabaticity of a unitary operation in a particular time interval given the available control resources. The interplay between adiabaticity, fidelity and robustness of the resulting pulses is examined for the case of single-qubit inversion, and superadiabatic pul…
Nonadiabatic holonomic quantum computation (NHQC) has been developed to shorten the construction times of geometric quantum gates. However, previous NHQC gates require the driving Hamiltonian to satisfy a set of rather restrictive conditions, reducing the robustness of the resulting geometric gates against control errors. Here we show that nonadiabatic geometric gates can be constructed in an extensible way, called NHQCþ, for maintaining both flexibility and robustness against certain types of noises. Consequently, this approach makes it possible to incorporate most of the existing optimal con…
It is shown that efficient and robust universal quantum computation is possible using the stimulated Raman adiabatic passage with a qubit chain as a pointer register.
High-fidelity and robust coherent population transfer is a major challenge in coherent quantum control. Different from the well known adiabatic condition, we present a rigorous adiabatic condition that is inspired by the idea of the Landau-Zener tunneling. Based on this, we propose a coherent population transfer approach, which just needs only one control parameter and depends on the eigenvalues of the systems. Compared to other approaches, such as fast quasiadiabatic dynamics, shortcut to adiabatic passage, we numerically demonstrate that our approach can provide a more high-fidelity and more…
We show a superpolynomial oracle separation between the power of adiabatic quantum computation with no sign problem and the power of classical computation.
Adiabatic quantum computation provides an alternative approach to quantum computation using a time-dependent Hamiltonian. The time evolution of entanglement during the adiabatic quantum search algorithm is studied, and its relevance as a resource is discussed.
We introduce high-order dynamical decoupling strategies for open system adiabatic quantum computation. Our numerical results demonstrate that a judicious choice of high-order dynamical decoupling method, in conjunction with an encoding which allows computation to proceed alongside decoupling, can dramatically enhance the fidelity of adiabatic quantum computation in spite of decoherence.
We prove the equivalence between adiabatic quantum computation and quantum computation in the circuit model. An explicit adiabatic computation procedure is given that generates a ground state from which the answer can be extracted. The amount of time needed is evaluated by computing the gap. We show that the procedure is computationally efficient.
The parametric deformations of quasienergies and eigenvectors of unitary operators are applied to the design of quantum adiabatic algorithms. The conventional, standard adiabatic quantum computation proceeds along eigenenergies of parameter-dependent Hamiltonians. By contrast, discrete adiabatic computation utilizes adiabatic passage along the quasienergies of parameter-dependent unitary operators. For example, such computation can be realized by a concatenation of parameterized quantum circuits, with an adiabatic though inevitably discrete change of the parameter. A design principle of adiaba…
Holonomic quantum computation: a scalable adiabatic architecture
本文提出一种基于全几何亚稳门的可扩展量子计算框架，通过微分几何分析揭示了这些门的几何本质和抗噪声特性，并结合最近的里德堡量子计算进展验证其可行性。
We propose a non-adiabatic scheme for geometric quantum computation with trapped ions. By making use of the Aharonov-Anandan phase, the proposed scheme not only preserves the globally geometric nature in quantum computation, but also provides the advantage of non-adiabaticity that overcomes the problem of slow evolution in the existing adiabatic schemes. Moreover, the present scheme requires only two atomic levels in each ion, making it an appealing candidate for quantum computation.
Adiabatic quantum computation, based on the adiabatic theorem, is a promising alternative to conventional quantum computation. The validity of an adiabatic algorithm depends on the existence of a nonzero energy gap between the ground and excited states. However, it is difficult to ascertain the exact value of the energy gap. In this paper, we put forward a theorem on the existence of nonzero energy gap for the Hamiltonians used in adiabatic quantum computation. It can help to effectively identify a large class of the Hamiltonians without energy-level crossing between the ground and excited sta…
Holonomic gates for quantum computation are commonly considered to be robust against certain kinds of parametric noise, the very motivation of this robustness being the geometric character of the transformation achieved in the adiabatic limit. On the other hand, the effects of decoherence are expected to become more and more relevant when the adiabatic limit is approached. Starting from the system described by Florio et al. [Phys. Rev. A 73, 022327 (2006)], here we discuss the behavior of non ideal holonomic gates at finite operational time, i.e., far before the adiabatic limit is reached. We …
This paper has been withdrawn as it has been superseded by 0808.2697
We analyze the computational power and limitations of the recently proposed 'quantum adiabatic evolution algorithm'.
Quantum adiabatic transfer is widely used in quantum computation and quantum simulation. However, the transfer speed is limited by the quantum adiabatic approximation condition, which hinders its application in quantum systems with a short decoherence time. Here we demonstrate quantum adiabatic state transfers that jump along geodesics in one-qubit and two-qubit superconducting transmons. This approach possesses the advantages of speed, robustness, and high fidelity compared with the usual adiabatic process. Our protocol provides feasible strategies for improving state manipulation and gate op…
For a finite XY chain and a finite two-dimensional Ising lattice, it is shown that the paramagnetic ground state is adiabatically transformed to the GHZ state in the ferromagnetic phase by slowly turning on the magnetic field. The fidelity between the GHZ state and an adiabatically evolved state shows a feature of the quantum phase transition.
We develop a non-adiabatic generalization of holonomic quantum computation in which high-speed universal quantum gates can be realized by using non-Abelian geometric phases. We show how a set of non-adiabatic holonomic one- and two-qubit gates can be implemented by utilizing optical transitions in a generic three-level $Λ$ configuration. Our scheme opens up for universal holonomic quantum computation on qubits characterized by short coherence times.
I show how to protect adiabatic quantum computation (AQC) against decoherence and certain control errors, using a hybrid methodology involving dynamical decoupling, subsystem and stabilizer codes, and energy gaps. Corresponding error bounds are derived. As an example I show how to perform decoherence-protected AQC against local noise using at most two-body interactions.
Quantum computation that combines the coherence stabilization virtues of decoherence-free subspaces and the fault tolerance of geometric holonomic control is of great practical importance. Some schemes of adiabatic holonomic quantum computation in decoherence-free subspaces have been proposed in the past few years. However, non-adiabatic holonomic quantum computation in decoherence-free subspaces, which avoids long run-time requirement but with all the robust advantages, remains an open problem. Here, we demonstrate how to realize non-adiabatic holonomic quantum computation in decoherence-free…
We propose a non-convex training objective for robust binary classification of data sets in which label noise is present. The design is guided by the intention of solving the resulting problem by adiabatic quantum optimization. Two requirements are imposed by the engineering constraints of existing quantum hardware: training problems are formulated as quadratic unconstrained binary optimization; and model parameters are represented as binary expansions of low bit-depth. In the present work we validate this approach by using a heuristic classical solver as a stand-in for quantum hardware. Testi…
We present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing.
We show that it is possible to use a classical computer to efficiently simulate the adiabatic evolution of a quantum system in one dimension with a constant spectral gap, starting the adiabatic evolution from a known initial product state. The proof relies on a recently proven area law for such systems, implying the existence of a good matrix product representation of the ground state, combined with an appropriate algorithm to update the matrix product state as the Hamiltonian is changed. This implies that adiabatic evolution with such Hamiltonians is not useful for universal quantum computati…
The proposal of the optical scheme for holonomic quantum computation is evaluated based on dynamical resolution to the system beyond adiabatic limitation. The time-dependent Schrödinger equation is exactly solved by virtue of the cranking representation and gauge transformation approach. Besides providing rigorous confirmation to holonomies of the geometrical prediction that holds for the ideally adiabatic situation, the dynamical resolution enables one to evaluate elaborately the amplitude of the nonadiabatic deviation, so that the errors induced to the quantum computation can be explicitly e…
Recent experiments show the existence of collective decoherence in quantum systems. We study the possibility of quantum computation in decoherence free subspace which is robust against such kind of decoherence processes. This passive protection protocol can be especially advantageous for continuous quantum computation such as quantum annealers. As an example we propose to use decoherence protected adiabatic quantum computation for the Grover search problem. Our proposal contains only two-body interactions, making it feasible in near-term quantum devices.
According to the quantum adiabatic theorem, we can in principle obtain a true vacuum of a quantum system starting from a trivial vacuum of a simple Hamiltonian. In actual adiabatic digital quantum simulation with finite time length and non-infinitesimal time steps, we can only obtain an approximate vacuum that is supposed to be a superposition of a true vacuum and excited states. We propose a procedure to improve the approximate vacuum.
Experimental demonstration of robustness under scaling errors for superadiabatic population transfer in a superconducting circuit
本文在超导transmon电路中实验演示了saSTIRAP布居转移对标度误差的鲁棒性，并发现高反绝热脉冲强度下出现超越常规saSTIRAP框架的新平台。
In quantum adiabatic algorithm, as the adiabatic parameter $s(t)$ changes slowly from zero to one with finite rate, a transition to excited states inevitably occurs and this induces an intrinsic computational error. We show that this computational error depends not only on the total computation time $T$ but also on the time derivatives of the adiabatic parameter $s(t)$ at the beginning and the end of evolution. Previous work (Phys. Rev. A \textbf{82}, 052305) also suggested this result. With six typical paths, we systematically demonstrate how to optimally design an adiabatic path to reduce th…
We examine the validity of the rotating wave approximation (RWA) in non-adiabatic holonomic single-qubit gates [New J. Phys. {\bf 14}, 103035 (2012)]. We demonstrate that the adoption of RWA may lead to a sharp decline in fidelity for rapid gate implementation and small energy separation between the excited and computational states. The validity of the RWA in the recent experimental realization [Nature (London) {\bf 496}, 482 (2013)] of non-adiabatic holonomic quantum computation for a superconducting qubit is examined.
The adiabatic quantum algorithm has drawn intense interest as a potential approach to accelerating optimization tasks using quantum computation. The algorithm is most naturally realised in systems which support Hamiltonian evolution, rather than discrete gates. We explore an alternative approach in which slowly varying measurements are used to mimic adiabatic evolution. We show that for certain Hamiltonians, which remain frustration-free all along the adiabatic path, the necessary measurements can be implemented through the measurement of random terms from the Hamiltonian. This offers a new, a…
Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyse a particular class of qauntum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum energy gap which governs the time required for a successful evolution is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is m…
We give a careful proof that a parallelized version of adiabatic quantum computation can efficiently simulate universal gate model quantum computation. The proof specifies an explicit parameter-dependent Hamiltonian $H(λ)$ that is based on ground state quantum computation [1]. We treat both a 1-dimensional configuration in which qubits on a line undergo nearest neighbor 2-qubit gates and an all-to-all configuration in which every qubit undergoes 2-qubit gates with every other qubit in the system.
We show enough evidence that a structured version of Adiabatic Quantum Computation (AQC) is efficient for most satisfiability problems. More precisely, when the success probability is fixed beforehand, the computational resources grow subexponentially in the number of qubits. Our study focuses on random satisfiability and exact cover problems, developing a multi-step algorithm that solves clauses one by one. Relating the computational cost to classical properties of the problem, we collect significant statistics with up to N=140 qubits, around the phase transitions, which is where the hardest …
We extend the work of Hen for the Bernstein-Vazirani problem and Simon's problem on an adiabatic quantum computer. Our results are based on the Ising objective function for quantum annealing. For each problem we determine its objective function, describe its Hamiltonian matrix, and show the Hamiltonian matrix for a small problem. Following the pattern of Hen, we discuss factoring of integers, particularly Shor's factoring algorithm in an adiabatic quantum computing environment.
We show an equivalence relation between fault-tolerant circuits for a stabilizer code and fault-tolerant adiabatic processes for holonomic quantum computation (HQC), in the case where quantum information is encoded in the degenerated ground space of the system Hamiltonian. By this equivalence, we can systematically construct a fault-tolerant HQC scheme, which can geometrically implement a universal set of encoded quantum gates by adiabatically deforming the system Hamiltonian. During this process, quantum information is protected from low weight thermal excitations by an energy gap that does n…
Adiabatic quantum computation starts from embedding a computational problem into a Hamiltonian whose ground state encodes the solution to the problem. This problem Hamiltonian, $H_{\rm p}$, is normally chosen to be diagonal in the computational basis, that is a product basis for qubits. We point out that $H_{\rm p}$ can be chosen to be non-diagonal in the computational basis. To be more precise, we show how to construct $H_{\rm p}$ in such a way that all its excited states are entangled with respect to the qubit tensor product structure, while the ground state is still of the product form and …
An explicit algorithm for the travelling salesman problem is constructed in the framework of adiabatic quantum computation, AQC. The initial Hamiltonian for the AQC process admits canonical coherent states as the ground state, and the target Hamiltonian has the shortest tour as the desirable ground state. Some estimates/bounds are also given for the computational complexity of the algorithm with particular emphasis on the required energy resources, besides the space and time complexity, for the physical process of (quantum) computation in general.
We indicate that there are points to keep in mind in utilizing quantum states prepared by the adiabatic quantum computation. Even if an instantaneous expectation value of a physical quantity for the adiabatically prepared quantum state is close to an expectation value for the true vacuum, this does not assure us that the prepared vacuum is close to the true vacuum. In general time average of the expectation value tend to systematically differ from the true value. Using a simple model we discuss how to diminish this systematic difference.
Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models provides a new vantage point from which to tackle the centra…
Extracting vacuum expectation values from approximate vacuum prepared by the adiabatic quantum computation
本研究提出一种从绝热量子计算制备的近似真空中提取真空期望值的方法，采用多层级结构的辅助量子比特提升精度，并通过经典模拟数字量子模拟器验证了(1+1)维Schwinger模型下的单、双量子比特系统效果。
We outline an efficient quantum-adiabatic algorithm that solves Simon's problem, in which one has to determine the `period', or xor-mask, of a given black-box function. We show that the proposed algorithm is exponentially faster than its classical counterpart and has the same complexity as the corresponding circuit-based algorithm. Together with other related studies, this result supports a conjecture that the complexity of adiabatic quantum computation is equivalent to the circuit-based computational model in a stronger sense than the well-known, proven polynomial equivalence between the two …
A cluster state cannot be a unique ground state of a two-body interacting Hamiltonian. Here, we propose the creation of a cluster state of logical qubits encoded in spin-1/2 particles by adiabatically weakening two-body interactions. The proposal is valid for any spatial dimensional cluster states. Errors induced by thermal fluctuations and adiabatic evolution within finite time can be eliminated ensuring fault-tolerant quantum computing schemes.
We propose a simple feedback-control scheme for adiabatic quantum computation with superconducting flux qubits. The proposed method makes use of existing on-chip hardware to monitor the ground-state curvature, which is then used to control the computation speed to maximize the success probability. We show that this scheme can provide a polynomial speed-up in performance and that it is possible to choose a suitable set of feedback-control parameters for an arbitrary problem Hamiltonian.
High-fidelity non-adiabatic dark state gates for neutral atoms
本文提出利用量子最优控制实现高保真的非绝热暗态门，以中性原子量子处理器中实现快速且鲁棒的双量子比特门操作。
Measurement-based quantum computation (MBQC) and holonomic quantum computation (HQC) are two very different computational methods. The computation in MBQC is driven by adaptive measurements executed in a particular order on a large entangled state. In contrast in HQC the system starts in the ground subspace of a Hamiltonian which is slowly changed such that a transformation occurs within the subspace. Following the approach of Bacon and Flammia, we show that any measurement-based quantum computation on a graph state with \emph{gflow} can be converted into an adiabatically driven holonomic comp…
We examine the adiabatic dynamics of a quantum system coupled to a noisy classical control field. A stochastic phase shift is shown to arise in the off-diagonal elements of the system's density matrix which can cause decoherence. We derive the condition for onset of decoherence, and identify the noise properties that drive decoherence. We show how this decoherence mechansim causes: (1) a dephasing of the observable consequences of the adiabatic geometric phase; and (2) the loss of computational efficiency of the Shor algorithm when run on a sufficiently noisy geometric quantum computer.
The NP-complete problem of the travelling salesman (TSP) is considered in the framework of quantum adiabatic computation (QAC). We first derive a remarkable lower bound for the computation time for adiabatic algorithms in general as a function of the energy involved in the computation. Energy, and not just time and space, must thus be considered in the evaluation of algorithm complexity, in perfect accordance with the understanding that all computation is physical. We then propose, with oracular Hamiltonians, new quantum adiabatic algorithms of which not only the lower bound in time but also t…
In previous implementations of adiabatic quantum algorithms using spin systems, the average Hamiltonian method with Trotter's formula was conventionally adopted to generate an effective instantaneous Hamiltonian that simulates an adiabatic passage. However, this approach had issues with the precision of the effective Hamiltonian and with the adiabaticity of the evolution. In order to address these, we here propose and experimentally demonstrate a novel scheme for adiabatic quantum computation by using the intrinsic Hamiltonian of a realistic spin system to represent the problem Hamiltonian whi…
We show that some composite pulses widely employed in NMR experiments are regarded as non-adiabatic geometric quantum gates with Aharanov-Anandan phases. Thus, we reveal the presence of a fundamental issue on quantum mechanics behind the traditional technique of the composite pulses. To examine the robustness of such composite pulses against fluctuations, we present a practical noise model in a two-level system. Then, we find that the composite pulses possesses purely geometrical nature even under a specific type of fluctuations.
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