The reliable and precise generation of quantum unitary transformations is essential to the realization of a number of fundamental objectives, such as quantum control and quantum information processing. Prior work has explored the optimal control problem of generating such unitary transformations as a surface optimization problem over the quantum control landscape, defined as a metric for realizing a desired unitary transformation as a function of the control variables. It was found that under the assumption of non-dissipative and controllable dynamics, the landscape topology is trap-free, implying that any reasonable optimization heuristic should be able to identify globally optimal solutions. The present work is a control landscape analysis incorporating specific constraints in the Hamiltonian corresponding to certain dynamical symmetries in the underlying physical system. It is found that the presence of such symmetries does not destroy the trap-free topology. These findings expand the class of quantum dynamical systems on which control problems are intrinsically amenable to solution by optimal control.
This work considers various families of quantum control landscapes (i.e. objective functions for optimal control) for obtaining target unitary transformations as the general solution of the controlled Schrödinger equation. We examine the critical point structure of the kinematic landscapes J_F (U) = ||(U-W)A||^2 and J_P (U) = ||A||^4 - |Tr(AA'W'U)|^2 defined on the unitary group U(H) of a finite-dimensional Hilbert space H. The parameter operator A in B(H) is allowed to be completely arbitrary, yielding an objective function that measures the difference in the actions of U and the target W on …
This paper considers the control landscape of quantum transitions in multi-qubit systems driven by unitary transformations with single-qubit interaction terms. The two-qubit case is fully analyzed to reveal the features of the landscape including the nature of the absolute maximum and minimum, the saddle points and the absence of traps. The results permit calculating the Schmidt state starting from an arbitrary two-qubit state following the local gradient flow. The analysis of multi-qubit systems is more challenging, but the generalized Schmidt states may also be located by following the local…
The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in $L^2(0,T)$, however, there are critical points for which the fidelity can assume any value in (0,1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows tha…
The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms; but (ii) the required time for convergence to optimal con…
The recent quantum information boom has effected a resurgence of interest in unitary coupled cluster (UCC) theory. Our group's interest in local energy landscapes of unitary ansätze prompted us to investigate the classical approach of truncating the Taylor series expansion (instead of a perturbative expansion) of UCCSD energy at second-order. This amounts to an approach where electron correlation energy is estimated by taking a single Newton-Raphson step from Hartree-Fock toward UCCSD. Such an approach has been explored previously, but the accuracy was not extensively studied. In this paper, w…
This work develops measures for quantifying the effects of field noise upon targeted unitary transformations. Robustness to noise is assessed in the framework of the quantum control landscape, which is the mapping from the control to the unitary transformation performance measure (quantum gate fidelity). Within that framework, a new geometric interpretation of stochastic noise effects naturally arises, where more robust optimal controls are associated with regions of small overlap between landscape curvature and the noise correlation function. Numerical simulations of this overlap in the conte…
The broad success of optimally controlling quantum systems with external fields has been attributed to the favorable topology of the underlying control landscape, where the landscape is the physical observable as a function of the controls. The control landscape can be shown to contain no suboptimal trapping extrema upon satisfaction of reasonable physical assumptions, but this topological analysis does not hold when significant constraints are placed on the control resources. This work employs simulations to explore the topology and features of the control landscape for pure-state population …
An estimate is derived for the volume fraction of a subset $C_ε^{P} = \{U : ||grad J(U)|\leq ε\}\subset\mathrm{U}(N)$ in the neighborhood of the critical set $C^{P}\simeq\mathrm{U}(\mathbf{n})P\mathrm{U}(\mathbf{m})$ of the kinematic quantum ensemble control landscape J(U) = Tr(UρU' O), where $U$ represents the unitary time evolution operator, ρ is the initial density matrix of the ensemble, and O is an observable operator. This estimate is based on the Hilbert-Schmidt geometry for the unitary group and a first-order approximation of $||grad J(U)||^2$. An upper bound on these near-critical vol…
We develop the embedded gradient vector field method, introduced in [8] and [9], for the case of the special unitary group $\mathcal{SU}(N)$ regarded as a constraint submanifold of the unitary group $\mathcal{U}(N)$. The optimization problem associated to the trace fidelity cost function defined on $\mathcal{SU}(N)$ that appears in the context of $\mathcal{SU}(N)$ quantum control landscapes is completely solved using the embedded gradient vector field method. We prove that for $N\geq 5$, the landscape is not $\mathcal{SU}(N)$-trap free, there are always kinematic local extrema that are not glo…
In optimal quantum control, control landscape phase transitions (CLPTs) indicate sharp changes occurring in the set of optimal protocols, as a physical model parameter is varied. Here, we demonstrate the existence of a new class of CLPTs, associated with changes in the topological properties of the optimal level set in a two-qubit state-preparation problem. In particular, the distance distribution of control protocols sampled through stochastic homotopic dynamics reveals discontinuous changes in the number of connected components in the optimal level set, as a function of the protocol duration…
Hybrid quantum-classical algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) are considered as one of the most encouraging approaches for taking advantage of near-term quantum computers in practical applications. Such algorithms are usually implemented in a variational form, combining a classical optimization method with a quantum machine to find good solutions to an optimization problem. The solution quality of QAOA depends to a high degree on the parameters chosen by the classical optimizer at each iteration. However, the solution landscape of those parameters is highly …
A quantum control landscape is defined as the expectation value of a target observable $Θ$ as a function of the control variables. In this work control landscapes for open quantum systems governed by Kraus map evolution are analyzed. Kraus maps are used as the controls transforming an initial density matrix $ρ_{\rm i}$ into a final density matrix to maximize the expectation value of the observable $Θ$. The absence of suboptimal local maxima for the relevant control landscapes is numerically illustrated. The dependence of the optimization search effort is analyzed in terms of the dimension of t…
The purpose of this paper is to introduce techniques of obtaining optimal ways to determine a d-level quantum state or distinguish such states. It entails designing constrained elementary measurements extracted from maximal abelian subsets of a unitary basis U for the operator algebra B(H) of a Hilbert space H of finite dimension d > 3 or, after choosing an orthonormal basis for H, for the *-algebra Md of complex matrices of order d > 3. Illustrations are given for the techniques. It is shown that the Schwinger basis U of unitary operators can give for d, a product of primes p and a, the ideal…
The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been shown to contain no critical point suboptimal traps, thereby enabling effective searches for fields that give the global maximum of the objective. This paper addresses the structure of the landscape as a complement to topological critical point features. Recent work showed that landscape structure is highly favorable for optimization of state-to-state transit…
Geometric Analysis of Variational Quantum Eigensolver
Michigan Institute for Computational Discovery and Engineering, Department of Electrical Engineering and Computer Science and Department of Statistics, University of Michigan（密歇根大学计算发现与工程研究所、电气工程和计算机科学系以及统计学系）
The broad success of theoretical and experimental quantum optimal control is intimately connected to the topology of the underlying control landscape. For several common quantum control goals, including the maximization of an observable expectation value, the landscape has been shown to lack local optima if three assumptions are satisfied: (i) the quantum system is controllable, (ii) the Jacobian of the map from the control field to the evolution operator is full-rank, and (iii) the control field is not constrained. In the case of the observable objective, this favorable analysis shows that th…
Energy Landscape Plummeting in Variational Quantum Eigensolver: Subspace Optimization, Non-iterative Corrections and Generator-informed Initialization for Improved Quantum Efficiency
Efficient optimization of quantum systems is a necessity for reaching fault tolerant thresholds. A standard tool for optimizing simulated quantum dynamics is the gradient-based \textsc{grape} algorithm, which has been successfully applied in a wide range of different branches of quantum physics. In this work, we derive and implement exact $2^{\mathrm{nd}}$ order analytical derivatives of the coherent dynamics and find improvements compared to the standard of optimizing with the approximate $2^{\mathrm{nd}}$ order \textsc{bfgs}. We demonstrate performance improvements for both the best and aver…
The contextual subspace variational quantum eigensolver (CS-VQE) is a hybrid quantum-classical algorithm that approximates the ground-state energy of a given qubit Hamiltonian. It achieves this by separating the Hamiltonian into contextual and noncontextual parts. The ground-state energy is approximated by classically solving the noncontextual problem, followed by solving the contextual problem using VQE, constrained by the noncontextual solution. In general, computation of the contextual correction needs fewer qubits and measurements compared with solving the full Hamiltonian via traditional …
We have developed in the previous works a statistical model of quantum fluctuation based on a chaotic deviation from infinitesimal stationary action which is constrained by the principle of Locality to have a unique exponential distribution up to a parameter that determines its average. The unitary Schrödinger time evolution with Born's statistical interpretation of wave function is recovered as a specific case when the average deviation from infinitesimal stationary action is given by $\hbar/2$ for all the time. This naturally suggests a possible generalization of the quantum dynamics and sta…
Massachusetts Institute of Technology（麻省理工学院）；University of Oxford（牛津大学）
We study the hardness of learning unitary transformations in $U(d)$ via gradient descent on time parameters of alternating operator sequences. We provide numerical evidence that, despite the non-convex nature of the loss landscape, gradient descent always converges to the target unitary when the sequence contains $d^2$ or more parameters. Rates of convergence indicate a "computational phase transition." With less than $d^2$ parameters, gradient descent converges to a sub-optimal solution, whereas with more than $d^2$ parameters, gradient descent converges exponentially to an optimal solution.
We investigate the energy-constrained (EC) diamond norm distance between unitary channels acting on possibly infinite-dimensional quantum systems, and establish a number of results. Firstly, we prove that optimal EC discrimination between two unitary channels does not require the use of any entanglement. Extending a result by Acín, we also show that a finite number of parallel queries suffices to achieve zero error discrimination even in this EC setting. Secondly, we employ EC diamond norms to study a novel type of quantum speed limits, which apply to pairs of quantum dynamical semigroups. We …
The channels, and more generally superoperators acting on the trace class operators of a quantum system naturally form a Banach space under the completely bounded trace norm (aka diamond norm). However, it is well-known that in infinite dimension, the norm topology is often "too strong" for reasonable applications. Here, we explore a recently introduced energy-constrained diamond norm on superoperators (subject to an energy bound on the input states). Our main motivation is the continuity of capacities and other entropic quantities of quantum channels, but we also present an application to the…
The fidelity and local unitary transformation are two widely useful notions in quantum physics. We study two constrained optimization problems in terms of the maximal and minimal fidelity between two bipartite quantum states undergoing local unitary dynamics. The problems are related to the geometric measure of entanglement and the distillability problem. We show that the problems can be reduced to semi-definite programming optimization problems. We give close-form formulae of the fidelity when the two states are both pure states, or a pure product state and the Werner state. We explain from t…
Quantum Fidelity Landscape-Guided Prior Calibration for Single-Circuit QGAN Image Generation
Shanghai Maritime University（上海海事大学）；Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局）；Singapore Institute of Technology（新加坡理工学院）；Singapore Management University (SMU)（新加坡管理大学）；Columbia University（哥伦比亚大学）；Tongji University（同济大学）
