We consider the optimal control problem of transferring population between states of a quantum system where the coupling proceeds only via intermediate states that are subject to decay. We pose the question whether it is generally possible to carry out this transfer. For a single intermediate decaying state, we recover the Stimulated Raman Adiabatic Passage (STIRAP) process which we identify as the global optimum in the limit of infinite control time. We also present analytic solutions for the case of transfer that has to proceed via two consecutive intermediate decaying states. We show that in this case, for finite power the optimal control does not approach perfect state transfer even in the infinite time limit. We generalize our findings to characterize the topologies of paths that can be achieved by coherent control under the assumption of finite power.
In a network of spin 1/2 particles, controlled through an external electro-magnetic field, the gyromagnetic ratio of each spin is a parameter that characterizes the interaction of the spin with the external control field. Multipartite networks are such that the spins are divided into subsets according to their gyromagnetic ratio and spins in one set interact in the same way with all spins in another set. Due to the presence of symmetries in this type of systems, the underlying Hilbert state space splits into invariant subspaces for the dynamics. Subspace controllability is verified if every un…
We consider a class of spin networks where each spin in a certain set interacts, via Ising coupling, with a set of central spins, and the control acts simultaneously on all the spins. This is a common situation for instance in NV centers in diamonds, and we focus on the physical case of up to two central spins. Due to the permutation symmetries of the network, the system is not globally controllable but it displays invariant subspaces of the underlying Hilbert space. The system is said to be subspace controllable if it is controllable on each of these subspaces. We characterize the given invar…
We consider the relation of symmetries and subspace controllability for spin networks with XXZ coupling subject to control of a single node by a local potential (Z-control). Such networks decompose into excitation subspaces. Focusing on the single excitation subspace it is shown that for single-node Z-controls external symmetries are characterized by eigenstates of the system Hamiltonian that have zero overlap with the control node, and there are no internal symmetries. It is further shown that there are symmetries that persist even in the presence of random perturbations. For uniformly couple…
The adiabatic theorem and "shortcuts to adiabaticity" for the adiabatic dynamics of time-dependent decoherence-free subspaces are explored in this paper. Starting from the definition of the dynamical stable decoherence-free subspaces, we show that, under a compact adiabatic condition, the quantum state follows time-dependent decoherence-free subspaces (the adiabatic decoherence free subspaces) into the target subspace with extremely high purity, even though the dynamics of the quantum system may be non-adiabatic. The adiabatic condition mentioned in the adiabatic theorem is very similar with t…
Symmetry is a fundamentally important concept in many branches of physics. In this work, we discuss two types of symmetries, external symmetry and internal symmetry, which appear frequently in controlled quantum spin chains and apply them to study various controllability problems. For spin chains under single local end control when external symmetries exists, we can rigorously prove that the system is controllable in each of the invariant subspaces for both XXZ and XYZ chains, but not for XX or Ising chains. Such results have direct applications in controlling antiferromagnetic Heisenberg chai…
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…
Non-adiabatic holonomic quantum gate in decoherence-free subspaces is of greatly practical importance due to its built-in fault tolerance, coherence stabilization virtues, and short run-time. Here we propose some compact schemes to implement two- and three-qubit controlled unitary quantum gates and Fredkin gate. For the controlled unitary quantum gates, the unitary operator acting on the target qubit is an arbitrary single-qubit gate operation. The controlled quantum gates can be directly implemented using non-adiabatic holonomy in decoherence-free subspaces and the required resource for the d…
A cavity QED implementation of the non-adiabatic holonomic quantum computation in decoherence-free subspaces is proposed with nitrogen-vacancy centers coupled commonly to the whispering-gallery mode of a microsphere cavity, where a universal set of quantum gates can be realized on the qubits. In our implementation, with the assistant of the appropriate driving fields, the quantum evolution is insensitive to the cavity field state, which is only virtually excited. The implemented non-adiabatic holonomies, utilizing optical transitions in the Λ type of three-level configuration of the nitrogen-v…
In the physics literature it is common to see the rotating wave approximation andthe adiabatic approximation used "in cascade" to justify the use of chirped pulses for two-level quantum systems driven by one external field, in particular when the resonance frequency of the system is not known precisely. Both approximations need relatively long time and are essentially based on averaging theory of dynamical systems. Unfortunately, the two approximations cannot be done independently since, in a sense, the two time scales interact. The purpose of this paper is to study how the cascade of the two …
A systematic perturbation scheme is developed for approximate solutions to the time-dependent Schroedinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band subspaces are suppressed to all orders and the dynamics operates in that subspace in terms of an effective intraband Hamiltonian. As novel applications we discuss the Born-Oppenheimer theory to second order and derive the nonperturbative defi…
We prove that a harmonic oscillator driven by Lindblad dynamics where the typical drive and loss channels are two-photon processes instead of single-photon ones, converges to a protected subspace spanned by two coherent states of opposite amplitude. We then characterize the slow dynamics induced by a perturbative single-photon loss on this protected subspace, by performing adiabatic elimination in the Lindbladian dynamics.
Adiabatic quantum computing (AQC) can be protected against thermal excitations via an encoding into error detecting codes, supplemented with an energy penalty formed from a sum of commuting Hamiltonian terms. Earlier work showed that it is possible to suppress the initial thermally induced excitation out of the encoded ground state, in the case of local Markovian environments, by using an energy penalty strength that grows only logarithmically in the system size, at a fixed temperature. The question of whether this result applies beyond the initial time was left open. Here we answer this in th…
Quantum annealing (QA) is a promising approach for not only solving combinatorial optimization problems but also simulating quantum many-body systems such as those in condensed matter physics. However, non-adiabatic transitions constitute a key challenge in QA. The choice of the drive Hamiltonian is known to affect the performance of QA because of the possible suppression of non-adiabatic transitions. Here, we propose the use of a drive Hamiltonian that preserves the symmetry of the problem Hamiltonian for more efficient QA. Owing to our choice of the drive Hamiltonian, the solution is searche…
By using transitionless quantum driving algorithm (TQDA), we present an efficient scheme for the shortcuts to the holonomic quantum computation (HQC). It works in decoherence-free subspace (DFS) and the adiabatic process can be speeded up in the shortest possible time. More interestingly, we give a physical implementation for our shortcuts to HQC with nitrogen-vacancy centers in diamonds dispersively coupled to a whispering-gallery mode microsphere cavity. It can be efficiently realized by controlling appropriately the frequencies of the external laser pulses. Also, our scheme has good scalabi…
We present a fast scheme for arbitrary unitary control of interacting bosonic atoms in a double-well. Assuming fixed inter-well tunnelling rate and intra-well interaction strength, we control the many-atom state by a discrete sequence of shifts of the single-well energies. For strong interactions, resonant tunnelling transitions implement beam-splitter U(2) rotations among atom number eigenstates, which can be combined and, thus, permit full controllability. By numerically optimizing such sequences of couplings at avoided level crossings (CALC), we extend the realm of full controllability to a…
We propose a method to adiabatically control an atomic ensemble using a decoherence-free subspace (DFS) within a dissipative cavity. We can engineer a specific eigenstate of the system's Lindblad jump operators by injecting a field into the cavity which deconstructively interferes with the emission amplitude of the ensemble. In contrast to previous adiabatic DFS proposals, our scheme creates a DFS in the presence of collective decoherence. We therefore have the ability to engineer states that have high multi-particle entanglements which may be exploited for quantum information science or metro…
Quantum phenomena of interest in connection with applications to computation and communication almost always involve generating specific transfers between eigenstates, and their linear superpositions. For some quantum systems, such as spin systems, the quantum evolution equation (the Schrödinger equation) is finite-dimensional and old results on controllability of systems defined on on Lie groups and quotient spaces provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, in an infinite-dimensional setting, controlling the evolution of quantum…
One often needs to estimate how fast an evolving state of a quantum system can depart from some target state or target subspace of a Hilbert space. Such estimates are known as quantum speed limits. We derive a quantum speed limit for a general time-dependent target subspace. When the target subspace is an instantaneous invariant subspace of a time-dependent Hamiltonian, the obtained quantum speed limit bounds the adiabatic fidelity, which is a figure of merit of quantum adiabaticity. We also compare two states evolving under two different Hamiltonians and derive a bound on the Loschmidt echo.
The quantum Zeno effect is recast in terms of an adiabatic theorem when the measurement is described as the dynamical coupling to another quantum system that plays the role of apparatus. A few significant examples are proposed and their practical relevance discussed. We also focus on decoherence-free subspaces.
We prove an analytical expression for the size of the gap between the ground and the first excited state of quantum adiabatic algorithm for the 3-satisfiability, where the initial Hamiltonian is a projector on the subspace complementary to the ground state. For large problem sizes the gap decreases exponentially and as a consequence the required running time is also exponential.
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.
We propose a decoherence-free subspaces (DFS) scheme to realize scalable quantum computation with trapped ions. The spin-dependent Coulomb interaction is exploited, and the universal set of unconventional geometric quantum gates is achieved in encoded subspaces that are immune from decoherence by collective dephasing. The scalability of the scheme for the ion array system is demonstrated, either by an adiabatic way of switching on and off the interactions, or by a fast gate scheme with comprehensive DFS encoding and noise decoupling techniques.
High-Precision Quantum Dynamics of He$_2$ over the b $^3Π_\mathrm{g}$-c $^3Σ_\mathrm{g}^+$ Electronic Subspace by including Non-adiabatic, Relativistic and QED Corrections and Couplings
The quantum mechanical motion of the atomic nuclei is considered over a single- or a multi-dimensional subspace of electronic states which is separated by a gap from the rest of the electronic spectrum over the relevant range of nuclear configurations. The electron-nucleus Hamiltonian is block-diagonalized up to $\mathcal{O}(\varepsilon^{n+1})$ through a unitary transformation of the electronic subspace and the corresponding $n$th-order effective Hamiltonian is derived for the quantum nuclear motion. Explicit but general formulae are given for the second- and the third-order corrections. As a …
Adiabatic evolution is used in a variety of quantum information processing tasks. However, the elimination of errors is not as well-developed as it is for circuit model processing. Here, we present a strategy to accelerate a reliable quantum adiabatic process by adding Leakage Elimination Operators (LEO) to the evolution which are a sequence of pulse controls acting in an adiabatic subspace. Using the Feshbach $PQ$ partitioning technique, we obtain an analytical solution which traces the footprint of the target eigenstate. The effectiveness of the LEO is independent of the specific form of the…
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. Remarkably this control scheme can be realized with net zero energy cost and it is fault-tolerant against fluctuation and noise, significantly relaxing the experimental constraints. We demonstrate how to realiz…
Optimal control theory is developed for the task of obtaining a primary objective in a subspace of the Hilbert space while avoiding other subspaces of the Hilbert space. The primary objective can be a state-to-state transition or a unitary transformation. A new optimization functional is introduced which leads to monotonic convergence of the algorithm. This approach becomes necessary for molecular systems subject to processes implying loss of coherence such as predissociation or ionization. In these subspaces controllability is hampered or even completely lost. Avoiding the lossy channels is a…
We revisit the protocols to create maximally entangled states between two Josephson junction (JJ) charge phase qubits coupled to a microwave field in a cavity as a quantum data bus. We devote to analyze a novel mechanism of quantum decoherence due to the adiabatic entanglement between qubits and the data bus, the off-resonance microwave field. We show that even through the variable of the data bus can be adiabatically eliminated, the entanglement between the qubits and data bus remains and can decoher the superposition of two-particle state. Fortunately we can construct a decoherence-free subs…
It is showed on the basis of the multiple-quantum operator algebra space formalism that ultra-broadband heteronuclear Hartmann-Hahn polarization transfer could be achieved by the amplitude- and frequency-modulation quasi-adiabatic excitation (90 degree) pulses, while it is usually difficult for the adiabatic inversion pulses to achieve effectively broadband Hartmann-Hahn transfer in a heteronuclear coupled two-spin system. The adiabatic and quasi-adiabatic pulses have an important property that within their activation bandwidth flip angle of the pulses is independent of the pulse duration and …
Quantum state manipulation with gates based on geometric phases acquired during cyclic operations promises inherent fault-tolerance and resilience to local fluctuations in the control parameters. Here we create a general non-Abelian and non-adiabatic holonomic gate acting in the $(\ket{0},\ket{2})$ subspace of a three-level transmon fabricated in a fully coplanar design. Experimentally, this is realized by simultaneously coupling the first two transitions by microwave pulses with amplitudes and phases defined such that the condition of parallel transport is fulfilled. We demonstrate the creati…
All known examples confirming the possibility of an exponential separation between classical simulation algorithms and stoquastic adiabatic quantum computing (AQC) exploit symmetries that constrain adiabatic dynamics to effective, symmetric subspaces. The symmetries produce large effective eigenvalue gaps, which in turn make adiabatic computation efficient. We present a classical algorithm to efficiently sample from the effective subspace of a $k$-local stoquastic Hamiltonian $H$, without a priori knowledge of its symmetries (or near-symmetries). Our algorithm maps any $k$-local Hamiltonian to…
Adiabatic or slowly varying gate operations are typically required in order to remain within the qubit subspace in an anharmonic oscillator. However significant speed ups are possible by using the two quadrature derivative-removal-by-adiabatic-gate(DRAG) technique, where a second time derivative pulse component burns a spectral hole near an unwanted transition. It is shown here, that simultaneous optimization of the detuning and the pulse norm in addition, further reduces leakage errors and significantly improve gate fidelities. However, with this optimization accounting for the AC Stark shift…
We propose a systematic and explicit method for the inverse engineering of the dynamics of an open quantum systems with no auxiliary Hamiltonian nor the prerequisite of adiabatic passage. In particular, we exploit the Lindblad dissipators in order to create a resource state or subspace of interest in the presence of decoherence. In a conceptual shift, the Lindblad dissipators, including multiple interactions that are central to determine the steady state in the long-time limit for an open quantum system, can be guided to produce a useful practical resource to achieve an arbitrary target state …
We provide two inequalities for estimating adiabatic fidelity in terms of two other more handily calculated quantities, i.e., generalized orthogonality catastrophe and quantum speed limit. As a result of considering a two-dimensional subspace spanned by the initial ground state and its orthogonal complement, our method leads to stronger bounds on adiabatic fidelity than those previously obtained. One of the two inequalities is nearly sharp when the system size is large, as illustrated using a driven Rice-Mele model, which represents a broad class of quantum many-body systems whose overlap of d…
We study the behavior of the non-adiabatic population transfer between resonances at an exceptional point in the spectrum of the hydrogen atom. It is known that, when the exceptional point is encircled, the system always ends up in the same state, independent of the initial occupation within the two-dimensional subspace spanned by the states coalescing at the exceptional point. We verify this behavior for a realistic quantum system, viz. the hydrogen atom in crossed electric and magnetic fields. It is also shown that the non-adiabatic hypothesis can be violated when resonances in the vicinity …
To implement a set of universal quantum logic gates based on non-Abelian geometric phases, it is a conventional wisdom that quantum systems beyond two levels are required, which is extremely difficult to fulfil for superconducting qubits, appearing to be a main reason why only single qubit gates was implemented in a recent experiment [A. A. Abdumalikov Jr \emph{et al}., Nature 496, 482 (2013)]. Here we propose to realize non-adiabatic holonomic quantum computation in decoherence-free subspace on circuit QED, where one can use only the two levels in transmon qubits, a usual interaction, and a m…
