RIGA (Reference Input Generation Algorithm) is a monotonic numerical method for generating quantum gates for closed systems that are described by Schrödinger equations. In a previous paper, the authors have presented a monotonic quantum gate generation algorithm, called here by L-RIGA (Lindblad-RIGA), that is able to consider open quantum systems described by Lindblad master equations. The authors have claimed in that paper, without proof, that L-RIGA was originally obtained from a version of RIGA. In this paper we present this version of RIGA, called here F-RIGA (Fock-RIGA) that can consider open quantum systems after converting them to the Fock-Liouville descripton. This conversion is based on the Fock-Map, that is, the map F that sends an × n Hermitian matrix into an 2-vector of a real Euclidean space. The contribution of this paper is to show that that L-RIGA and F-RIGA are equivalent in the sense that, for each step ℓ, the data that is obtained by F-RIGA is transformed by the inverse of the Fock-Map into the data that is obtained in the same step of L-RIGA, letting the corresponding Lyapunov functions invariant. Furthermore, thanks to the great similarity between L-RIGA and a version of the Krotov method, a subproduct of this work is also establish a strong connection between that version of the Krotov method and the family of algorithms that are called by RIGA.
RIGA (Reference Input Generation Algorithm) is a monotonic numerical method for generating quantum gates for closed systems that are described by Schrödinger equations. In a previous paper, the authors have presented a monotonic quantum gate generation algorithm, called here by L-RIGA (Lindblad-RIGA), that is able to consider open quantum systems described by Lindblad master equations. The authors have claimed in that paper, without proof, that L-RIGA was originally obtained from a version of RIGA. In this paper we present this version of RIGA, called here F-RIGA (Fock-RIGA) that can consider open quantum systems after converting them to the Fock-Liouville descripton. This conversion is based on the Fock-Map, that is, the map F that sends an × n Hermitian matrix into an 2-vector of a real Euclidean space. The contribution of this paper is to show that that L-RIGA and F-RIGA are equivalent in the sense that, for each step ℓ, the data that is obtained by F-RIGA is transformed by the inverse of the Fock-Map into the data that is obtained in the same step of L-RIGA, letting the corresponding Lyapunov functions invariant. Furthermore, thanks to the great similarity between L-RIGA and a version of the Krotov method, a subproduct of this work is also establish a strong connection between that version of the Krotov method and the family of algorithms that are called by RIGA.
A new approach called Reference Input Generation Algorithm (RIGA) is introduced for quantum gate generation. Let X¯ℓ−1(t) be the trajectory that is obtained in step ℓ−1, with X¯ℓ−1(t)=I. In step ℓ, X¯ℓ−1 is right-translated in order to displace X¯ℓ−1(Tf) to Xgoal. The translated trajectory is used as a reference for a Lyapunov-based tracking control law, generating X¯ℓ(T), and so on. A proof of the exponential convergence of X¯ℓ−1(Tf) to zero is provided for Tf large enough. Two examples present numerical experiments regarding N coupled qubits. The first example considers N = 3 with a known minimum time T∗. It presents excellent results for Tf=T∗. The second example is a benchmark for the comparison between RIGA and GRAPE, considering a Hadamard gate for the systems with N = 2, 3, …, 10 qubits. The runtime of RIGA could be improved, and GRAPE was implemented in a faster CPU. However, RIGA presents results that are similar to GRAPE, with faster runtime in some cases, showing that the RIGA is indeed a promising algorithm.
We present a monotonic numerical algorithm including time optimization for generating quantum gates for open systems. Such systems are assumed to be governed by Lindblad master equations for the density operators on a large Hilbert-space whereas the quantum gates are relative to a sub-space of small dimension. Starting from an initial seed of the control input, this algorithm consists in the repetition of the following two steps producing a new control input: (A) backwards integration of adjoint Lindblad-Master equations (in the Heisenberg-picture) from a set of final conditions encoding the quantum gate to generate; (B) forward integration of Lindblad-Master equations in closed-loop where a Lyapunov based control produced the new control input. The numerical stability is ensured by the stability of both the open-loop adjoint backward system and the forward closed-loop system. A clock-control input can be added to the usual control input. The obtained monotonic algorithm allows then to optimise not only the shape of the control input, but also the gate time. Preliminary numerical implementations indicate that this algorithm is well suited for cat-qubit gates, where Hilbert-space dimensions (2 for the Z-gate and 4 for the CNOT-gate) are much smaller than the dimension of the physical Hilbert-space involving mainly Fock-states (typically 20 or larger for a single cat-qubit). This monotonic algorithm, based on Lyapunov control techniques, is shown to have a straightforward interpretation in terms of optimal control: its stationary conditions coincides with the first-order optimality conditions for a cost depending linearly on the final values of the quantum states.
The problem of encoded quantum gate generation is studied in this paper. The idea is to consider a quantum system of higher dimension n than the dimension nˉ of the quantum gate to be synthesized. Given two orthonormal subsets E={e1,e2,…,enˉ} and F={f1,f2,…,fnˉ} of Cn, the problem of encoded quantum gate generation consists in obtaining an open loop control law defined in an interval [0,Tf] in a way that all initial states ei are steered to exp(ϕ)fi,i=1,2,…,nˉ up to some desired precision and to some global phase ϕ∈R. This problem includes the classical (full) quantum gate generation problem, when nˉ=n, the state preparation problem, when nˉ=1, and finally the encoded gate generation when $ 1 < \bar n < n$. Hence, three problems are unified here within a unique common approach. The \emph{Reference Input Generation Algorithm (RIGA)} is generalized in this work for considering the encoded gate generation problem for closed quantum systems. A suitable Lyapunov function is derived from the orthogonal projector on the support of the encoded gate. Three case-studies of physical interest indicate the potential interest of such numerical algorithm: two coupled transmon-qubits, a cavity mode coupled to a transmon-qubit, and a chain of N qubits, including a large dimensional case for which N=10.
We introduce Liouville-Fock-state lattices (LFSLs) as a framework for visualizing open quantum systems through matrix representations of the Lindblad master equation. These generalize Hamiltonian Fock-state lattices, which have been considered for realizing synthetic dimensions in quantum simulators. By vectorizing the Lindblad master equation, the dynamics map to a doubled Hilbert space, naturally forming a synthetic lattice. Unlike the unitary evolution of pure states, LFSLs feature non-Hermitian dynamics with drifts, sources, and sinks—paralleling stochastic classical lattices. As such, we suggest that these systems can serve as potential quantum simulators of classical lattice models, with a computational advantage arising from large state spaces encoded in quantum models. For this purpose, we analyze both the Fock representation and alternative positive-semidefinite representations that yield real, non-negative populations, akin to classical probability distributions. Within this formalism, we also demonstrate a phenomenon, which we term environment-induced frustration, in which competing terms in the LFSL give rise to geometric frustration and a massively degenerate steady-state manifold.
Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) dtdρt=Lρt, where L is a linear (super-)operator and ρt is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state ρ∞ (such that Lρ∞=0), the first few excited states, and trajectories t↦ρt. In finite dimension, ρt is an n×n matrix, L thus typically costs n4 to store explicitly as a dense matrix, and O(n6) to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, L usually costs only O(n3) to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution S, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map Φ whose fixed point is directly related to ρ∞, all the other eigenvalues having smaller magnitude. The map Φ is thus well suited to iterative methods, and ρ∞ can be found in a few Arnoldi iterations. Using the same inverse map S−1 as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like O(n3) per iteration and offer state-of-the-art performance on CPU and GPU.
This paper considers right-invariant and controllable quantum systems with m inputs u = (u1,… , um) and state X(t) evolving on the unitary Lie group U(n). The ε-steering problem is introduced and solved for systems with drift: given any initial condition X0 at the initial time instant t0 ≥ 0, any goal state Xgoal∈ U(n) and ε > 0, find a control law such that dist(X(t0+t‾),Xgoal)≤ϵ, where t‾>0 is big enough and dist(X1, X2) is a convenient right-invariant notion of distance between two elements X1, X2 ∈ U(n). The purpose is to approximately generate arbitrary quantum gates corresponding to Xgoal. This is achieved by solving a tracking problem for a special kind of reference trajectories W‾: [t0, ∞) → U(n), which are here called c-universal reference trajectories. It is shown that, for this special kind of trajectories, the tracking problem can be solved up to an error ε for any reference trajectory X‾(t)=W‾(t)R which is a right-translation of W‾(t), at least when dist(X(t0),X‾(t0)) is finite. Furthermore, it is shown that dist(X(t),X‾(t)) converges uniformly exponentially to zero in the sense that the rate of convergence is independent of t0, R and X0. The approach considered here for showing such convergence is a generalisation of the results of a previous paper of the authors, which is mainly based on the central ideas of Jurdjevic and Quinn and Coron's return method. Taking a right-translation R such that X‾(t0+t‾)=Xgoal, one may solve the ε-steering problem by solving the tracking problem for the reference trajectory X‾(t), at least when dist(X(t0),X‾(t0))≤c. When dist(X(t0),X‾(t0))≥c, it is shown that the ε-steering problem can be globally solved in a two-iteration procedure. The underlying algorithmic complexity to get the steering control is essentially equivalent to the numerical integration of the Cauchy problem governing X(t). A numerical example considering a Toffoli quantum gate on U(8) for a chain of three coupled qubits that are controlled only locally is presented.