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arXiv:0709.4484 · Physical Review A

Time-optimal synthesis of unitary transformations in a coupled fast and slow qubit system

Robert Zeier, Haidong Yuan, Navin Khaneja

Published Mar 20, 2008• 28 citations
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Abstract

In this paper, we study time-optimal control problems related to a system of two coupled qubits where the time scales involved in performing unitary transformations on each qubit are significantly different. In particular, we address the case where unitary transformations produced by evolutions of the coupling take a much longer time as compared to the time required to produce unitary transformations on the first qubit, but a much shorter time as compared to the time to produce unitary transformations on the second qubit. We present a canonical decomposition of SU(4) in terms of the subgroup $\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$, which is natural in understanding the time-optimal control problem of such a coupled qubit system with significantly different time scales. A typical setting involves dynamics of a coupled electron-nuclear spin system in pulsed electron paramagnetic resonance experiments at high fields. Using the proposed canonical decomposition, we give time-optimal control algorithms to synthesize various unitary transformations of interest in coherent spectroscopy and quantum-information processing.

DOI
10.1103/physreva.77.032332

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In this paper, we study time-optimal control problems related to a system of two coupled qubits where the time scales involved in performing unitary transformations on each qubit are significantly different. In particular, we address the case where unitary transformations produced by evolutions of the coupling take a much longer time as compared to the time required to produce unitary transformations on the first qubit, but a much shorter time as compared to the time to produce unitary transformations on the second qubit. We present a canonical decomposition of SU(4) in terms of the subgroup SU(2)\ifmmode×\else\texttimes\fiSU(2)\ifmmode×\else\texttimes\fiU(1)\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)\ifmmode\times\else\texttimes\fi{}\text{U}(1)SU(2)\ifmmode×\else\texttimes\fiSU(2)\ifmmode×\else\texttimes\fiU(1), which is natural in understanding the time-optimal control problem of such a coupled qubit system with significantly different time scales. A typical setting involves dynamics of a coupled electron-nuclear spin system in pulsed electron paramagnetic resonance experiments at high fields. Using the proposed canonical decomposition, we give time-optimal control algorithms to synthesize various unitary transformations of interest in coherent spectroscopy and quantum-information processing.
In this paper, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. The ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of evolution, $\g$, are decomposed into fast generators $\k$ (local Hamiltonians) and slow generators $\p$ (couplings) as a Cartan decomposition $\g = \p \oplus \k$. Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. In this paper, we carry out a second order analysis of time optimality.
In this chapter, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation.Time optimal control helps to minimize relaxation losses.In a two qubit system, the ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems.The generators of unitary evolution, g, are decomposed into fast generators k (local Hamiltonians) and slow generators p (couplings) as a Cartan decomposition g ¼ p ⊕ k.Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems.The main contribution of the chapter is, we carry out a global analysis of time optimality.
Experiments in coherent nuclear and electron magnetic resonance and quantum computing in general correspond to control of quantum-mechanical systems, guiding them from initial to final target states by unitary transformations. The control inputs (pulse sequences) that accomplish these unitary transformations should take as little time as possible so as to minimize the effects of relaxation and decoherence and to optimize the sensitivity of the experiments. Here, we derive a time-optimal sequences as fundamental building blocks to synthesize unitary transformations. Such sequences can be widely implemented on various physical systems, including the simulation of effective Hamiltonians for topological quantum computing on spin lattices. Experimental demonstrations are provided for a system consisting of three nuclear spins.
In this paper, we study the design of pulse sequences for nuclear magnetic resonance spectroscopy as a problem of time optimal control of the unitary propagator. Radio-frequency pulses are used in coherent spectroscopy to implement a unitary transfer between states. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group. What is the minimum time required to steer the system from some initial point to a specified final point? In nuclear magnetic resonance (NMR) spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two-spin system.
The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time-dependent Hamiltonian. The inverse problem of finding the field that generates a specific unitary transformation is the subject of study. The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space. Optimal control theory is used to solve the inversion problem irrespective of the initial input state. A unified formalism based on the Krotov method is developed leading to a different scheme. The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ electronic state of Na2.{\mathrm{Na}}_{2}.Na2​. Raman-like transitions through the $A{}^{1}{\ensuremath{\Sigma}}_{u}^{+}$ electronic state induce the transitions. Light fields are found that are able to implement the Fourier transform within a picosecond time scale. Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse. Of the schemes studied, the square modulus scheme converges fastest. A study of the implementation of the Q qubit Fourier transform in the Na2{\mathrm{Na}}_{2}Na2​ molecule was carried out for up to five qubits. The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels. The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized.
In this paper, we give a characterization of all the unitary transformations that can be synthesized in a given time for a two-qubit system in the presence of general time varying coupling tensor. This characterization helps to compute the minimum time and the shortest pulse sequence for generating a general two-qubit transformation under nonstationary interactions. The methods presented here can be applied in design of time optimal pulse sequences for transferring coherence and polarization between coupled spins with time varying couplings as in solid-state NMR under magic angle spinning.