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arXiv:quant-ph/0608039 · Physical Review A

Time-optimal unitary operations

A. Carlini, Akio Hosoya, Tatsuhiko Koike, Yosuke Okudaira

Published Apr 10, 2007• 126 citations
arXivHTMLPDF

Abstract

Extending our previous work on time-optimal quantum state evolution [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time-optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time-optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time-optimal unitary evolution for three examples of target quantum gates, namely, the swap of qubits, the quantum Fourier transform, and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.

DOI
10.1103/physreva.75.042308

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Extending our previous work on time-optimal quantum state evolution [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time-optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time-optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time-optimal unitary evolution for three examples of target quantum gates, namely, the swap of qubits, the quantum Fourier transform, and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.
We present a general framework for finding the time-optimal quantum evolution inspired by the classical problem called the brachistochrone problem.At first, the time-optimal control problem for achieving a pure state evolution from a given initial state to a desired final state is formulated.We minimize the duration time for this transition by finding the optimal control for the Hamiltonian in an available set subject to a certain constraint dictated by physical conditions.Then, it will be extended to the unitary evolution which has greater relevance for quantum computation.This provides a systematic way to make an efficient unitary gate on a certain physical system.To demonstrate this framework, we choose the anisotropic Heisenberg model and find a general solution.Since it is not always possible to find a general solution for the fundamental equation for the time-optimal control, we introduce the simulated annealing method as a simple numerical method to show that our framework can be used for the system whose general solution cannot be found.The numerical analysis based on the Krotov iteration method is introduced and applied to the time-optimal quantum control problem to solve the problem efficiently.The optimal realization of the quantum Fourier transform for two or more qubit systems are derived.We consider two generalizations of the quantum brachistochrone for mixed state evolutions.One is the unitary evolution with ancilla system ended up with the final measurement on the ancilla.The other is the repetition of measurement or decoherence model which can be represented by the evolution with the master equation in the Lindblad form.Also, the relation between the fidelity to the target and the duration time is considerd.A.4 Quantum speed limit . . . . . . . . .
We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.
A quantum gate is realized by specific unitary transformations operating on states representing qubits. Considering a quantum system employed as an element in a quantum computing scheme, the task is therefore to enforce the pre-specified unitary transformation. This task is carried out by an external time dependent field. Optimal control theory has been suggested as a method to compute the external field which alters the evolution of the system such that it performs the desire unitary transformation. This study compares two recent implementations of optimal control theory to find the field that induces a quantum gate. The first approach is based on the equation of motion of the unitary transformation. The second approach generalizes the state to state formulation of optimal control theory. This work highlight the formal relation between the two approaches.
An important result from the mid nineties shows that any unitary evolution may be realized as a sequence of controlled-not and one-qubit gates. This work surveys especially efficient circuits in this library, in the special case of evolutions on two-quantum bits. In particular, we show that to construct an arbitrary two-qubit state from |00>, one CNOT gate suffices. To simulate an arbitrary two-qubit operator up to relative phases, two CNOTs suffice. To simulate an arbitrary two-qubit operator up to global phase, three CNOTs suffice. In each case, we construct an explicit circuit and prove optimality in the generic case. We also contribute a procedure to determine the minimal number of CNOT gates necessary to simulate a given two-qubit operator up to global phase. We use this procedure to discuss timing a given Hamiltonian to simulate the CNOT and to determine an optimal circuit for the two-qubit Quantum Fourier Transform. Our constructive proofs amount to circuit synthesis algorithms and have been coded in C++.
The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it. A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties. Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement. Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output. In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.
Experimental quantum simulation of a Hamiltonian HHH requires unitary operator decomposition (UOD) of its evolution unitary $U=\text{exp}(\ensuremath{-}iHt)$ in terms of native unitary operators of the experimental system. Here, using a genetic algorithm, we numerically evaluate the most generic UOD (valid over a continuous range of Hamiltonian parameters) of the unitary operator U,U,U, termed fidelity-profile optimization. The optimization is obtained by systematically evaluating the functional dependence of experimental unitary operators (such as single-qubit rotations and time-evolution unitaries of the system interactions) to the Hamiltonian (HHH) parameters. Using this technique, we have solved the experimental unitary decomposition of a controlled-phase gate (for any phase value), the evolution unitary of the Heisenberg XY interaction, and simulation of the Dzyaloshinskii-Moriya (DM) interaction in the presence of the Heisenberg XY interaction. Using these decompositions, we studied the entanglement dynamics of a Bell state in the DM interaction and experimentally verified the entanglement preservation procedure of Hou et al. [Ann. Phys. (N.Y.) 327, 292 (2012)] in a nuclear magnetic resonance quantum information processor.