Holonomic quantum computation offers a promising route to robust quantum gates, but decoherence remains a central obstacle in realistic implementations. Here we develop a nonadiabatic holonomic scheme for a driven three-level system in the no-jump regime described by an effective non-Hermitian Hamiltonian. Within a biorthogonal framework, tailored complex pulses enforce exact closure of the computational-subspace evolution at the final time despite the underlying nonunitary dynamics, enabling arbitrary single-qubit holonomic gates without requiring cyclic evolution in its orthogonal complement. In contrast to existing non-Hermitian treatments, which either neglect the overall exponential prefactor or, in adiabatic settings, include dissipation only on the auxiliary excited level, our scheme incorporates decay and dephasing of all bare eigenstates directly into the pulse design, so that dissipation does not reduce the no-jump gate fidelity.
Implementing quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators, is a particularly promising paradigm for the design of intrinsically stable quantum computers [1]. In contrast to dynamic phases, the geometric phase accumulated by a quantum system propagating through a Hilbert space
Hdepends exclusively on its path. In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of multi-dimensional, non-Abelian geometric phases - so called holonomies [2]. Anandan later dropped the requirement of adiabaticity to create holonomies, that are truly time-independent [3]. Non-adiabatic holonomies rely on a subspace
Hgeoof the Hilbert-space that is spanned by states
{∣Φk⟩}kthat fulfill
(Φk∣H^∣Φj⟩=0, where
H^is the system's Hamiltonian. Restricting the propagation to
Hgeoensures parallel transport and, thus, a purely geometric phase (see Fig. 1a) [4], [5]. Quantum optics constitutes a particularly versatile platform for quantum information processing, and in particular for the construction of non-adiabatic holonomic quantum computers: In addition to integration and miniaturization provided by the platform, the bosonic nature of photons also conveniently allows for multiple excitations of the same mode, readily expanding
Hgeoand enabling the synthesis of holonomies from higher symmetry groups
U(N)as larger and more capable computational units [6], [7].
Quantum computation is much more effective than classical computation in solving many problems such as factoring large integers and searching unsorted databases. However, such effectiveness relies on the ability to perform universal high-fidelity quantum gates. One main challenge in achieving such high-fidelity gates is to reduce control errors of a quantum system. To overcome this problem, various proposals of fault-tolerant quantum computation are proposed. A promising one of such proposals is nonadiabatic holonomic quantum computation. Nonadiabatic holonomic quantum computation is realized by using a quantum system with a subspace satisfying both the cyclic evolution and parallel transport conditions. For an N dimensional quantum system with Hamiltonian H(t) and evolution operator U(t)=Texp[−i∫0tH(t')dt'] , if there exists a time-dependent L dimensional subspace S(t) spanned by the orthonormal vectors {|ϕk(t)〉=U(t)|ϕk(0)〉}k=1L that satisfy the two conditions: (1) ∑k=1L|ϕk(τ)〉〈ϕk(τ)|= ∑k=1L|ϕk(0)〉〈ϕk(0)| , with τ being the evolution period, and (2) 〈ϕk(t)|H(t)|ϕl(t)〉=0, k,l=1,2,⋯,L, then the unitary transformation U(τ) is a holonomic gate on the L dimensional subspace S(0) spanned by {|ϕk(0)〉}k=1L . The work of realizing nonadiabatic holonomic gate is to build a physical system with a Hamiltonian satisfying both the cyclic evolution condition and the parallel transport condition. Nonadiabatic holonomic quantum computation is based on nonadiabatic non-Abelian geometric phases. Since nonadiabatic non-Abelian geometric phases are only dependent on evolution paths but independent of evolution details, nonadiabatic holonomic gates are robust against control errors. Besides, nonadiabatic holonomic gates do not require the long run-time evolution that is necessary for adiabatic holonomic gates. Due to the merits of both robustness against control errors and high-speed realization, nonadiabatic holonomic quantum computation has received increasing attention in theory and experiment. Various schemes of nonadiabatic holonomic quantum computation have been proposed based on different physical systems, and a number of these schemes have been experimentally demonstrated with nuclear magnetic resonance systems, nitrogen-vacancy center systems, and superconducting systems. The progress of nonadiabatic holonomic quantum computation is also reflected in the improvement of the methods of designing quantum gates. The single-shot scheme, single-loop multiple-pulse scheme and composite gate scheme of nonadiabatic quantum gate are based on the improved methods. In particular, the single-shot scheme simplifies the operation of the one-qubit gate, so that the operation of any one-qubit gate, which needs to be realized by combining two non-commuting π-rotation gate in the original scheme, can be simply realized by a single-shot operation. In this paper, we propose a scheme of nonadiabatic holonomic quantum computation with atom-cavity system. The universal set of quantum gates in our scheme consists of an arbitrary one-qubit nonadiabatic holonomic gate and a two-qubit nonadiabatic holonomic controlled-PHASE gate. To realize the arbitrary one-qubit gate, we use two resonant laser pulses and an off-resonant laser pulse to drive a single four-level atom. With the aid of Stark shifts, the one-qubit gate can be performed in one-step only by changing the Rabi frequencies and the phases of the laser pulses. To realize the two-qubit controlled-PHASE gate, we use two off-resonant laser pulses to drive the two four-level atoms in the cavity. The transitions between the two-atom ground state and the two-atom auxiliary state is facilitated by exchanging virtual photons through a common cavity mode. In this process, the effective two-atom transition is disentangled from the cavity mode and therefore the two-qubit gate is insensitive to the cavity decay.
As a paradigm for describing open-quantum-system dynamics, the non-Hermitian Schrödinger equation generally modifies state norms, complicating coherent control over the system's Hilbert space. With a general instantaneous frame spanned by biorthogonal ancillary bases, we develop a biorthogonal quantum-control theory for discrete-variable systems governed by time-dependent non-Hermitian Hamiltonian. In particular, we derive a necessary and sufficient condition for extracting the transitionless pathways that transport parallel in ket and bra states. When this condition is satisfied by a set of projectors constructed with biorthogonal basis states, the non-Hermitian Hamiltonian becomes diagonal in the relevant biorthogonal subspace, enabling arbitrary initial state to evolve as a superposition of controlled transitionless passages. We further identify the parametric constraints under which the transformation linking the initial and final states becomes unitary. As a result, the final system state is automatically normalized even though the transitional dynamics is nonunitary. Our theory is justified by the bidirectional population transfer and a holomonic quantum gate implemented by a non-Hermitian three-level system in
Λ type. For the population transfer, controlled gain and loss reduce the required coherent pulse area, enabling shorter transfer times than the Hermitian benchmark with the same average Rabi frequency. Our work integrates the universal quantum control (UQC) into the biorthogonal representation, displaying full-rank passage engineering with coherent operations in non-Hermitian systems.
In the holonomic approach to quantum computation, information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and manipulated by the associated holonomic gates. These are realized in terms of the non-Abelian Berry connection and are obtained by driving the control parameters along adiabatic loops. We show how it is possible for a specific model to explicitly determine the loops generating any desired logical gate, thus producing a universal set of unitary transformations. In a multipartite system unitary transformations can be implemented efficiently by sequences of local holonomic gates. Moreover, a conceptual scheme for obtaining the required Hamiltonian family, based on frequently repeated pulses, is discussed, together with a possible process whereby the initial state can be prepared and the final one can be measured.