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Physical Review A

Shortest paths for efficient control of indirectly coupled qubits

Navin Khaneja, Björn Heitmann, Andreas K. Spörl, Haidong Yuan, Thomas Schulte‐Herbrüggen, Steffen J. Glaser

Published Jan 22, 2007• 63 citations
Publisher

Abstract

What is the time-optimal way of realizing quantum operations? Here, we show how important instances of this problem can be related to the study of shortest paths on the surface of a sphere under a special metric. Specifically, we provide an efficient synthesis of a controlled-NOT (CNOT) gate between qubits (spins $\frac{1}{2}$) coupled indirectly via Ising-type couplings to a third spin. Our implementation of the CNOT gate is significantly shorter than conventional approaches. The pulse sequences for efficient manipulation of our coupled spin system are obtained by explicit computation of geodesics on a sphere under the special metric. These methods are also used for the efficient synthesis of indirect couplings and of the Toffoli gate. We provide experimental realizations of the presented methods on a linear three-spin chain with Ising couplings.

DOI
10.1103/physreva.75.012322

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What is the time-optimal way of realizing quantum operations? Here, we show how important instances of this problem can be related to the study of shortest paths on the surface of a sphere under a special metric. Specifically, we provide an efficient synthesis of a controlled-NOT (CNOT) gate between qubits (spins 12\frac{1}{2}21​) coupled indirectly via Ising-type couplings to a third spin. Our implementation of the CNOT gate is significantly shorter than conventional approaches. The pulse sequences for efficient manipulation of our coupled spin system are obtained by explicit computation of geodesics on a sphere under the special metric. These methods are also used for the efficient synthesis of indirect couplings and of the Toffoli gate. We provide experimental realizations of the presented methods on a linear three-spin chain with Ising couplings.
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In this article, we study optimal control of dynamics of a linear chain of three spin 1∕21∕21∕2, weakly coupled with unequal Ising couplings. We address the problem of time-optimal synthesis of multiple spin quantum coherences. We derive time-optimal pulse sequences for creating desired spin order by computing geodesics on a sphere under a special metric. The solution to the geodesic equation is related to the nonlinear oscillator equation and the minimum time to create multiple spin order can be expressed in terms of an elliptic integral. These techniques are used for efficient creation of multiple spin coherences in Ising spin chains with unequal couplings.
We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with a symmetric coupling J plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the CNOT(1, 3) between the indirectly coupled qubits 1 and 3 is , i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh–Hadamard rotation in the Hilbert space of qubit 3 shows that the time-optimal synthesis of the CNOT ± (1, 3) (which acts as the identity when the control qubit 1 is in the state |0⟩, while if the control qubit is in the state |1⟩, the target qubit 3 is flipped as | ± ⟩ → |∓⟩) also requires the same time T .
Experiments in coherent nuclear and electron magnetic resonance, and optical spectroscopy correspond to control of quantum mechanical ensembles, 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 give efficient syntheses of various unitary transformations on Ising spin chains of arbitrary length. The efficient realization of the unitary transformations presented here is obtained by computing geodesics on a sphere under a special metric. We show that contrary to the conventional belief, it is possible to propagate a spin order along an Ising spin chain with coupling strength JJJ (in units of Hz), significantly faster than ${(2J)}^{\ensuremath{-}1}$ per step. The methods presented here are expected to be useful for immediate and future applications involving control of spin dynamics in coherent spectroscopy and quantum information processing.
We propose a protocol for implementing Barenco-type multi-qubit controlled gates using short driven spin chains. Starting from an Ising interaction with a transverse drive on the last spin, we construct an effective two-qubit Hamiltonian whose time evolution implements the Barenco gate V2(φ,ω,φ)V_2(\varphi,ω,φ)V2​(φ,ω,φ) and, in particular, a CNOT gate. We then embed this construction into a three-qubit XXZXXZXXZ chain to realize the three-qubit Barenco gate V3(φ,ω,φ)V_3(\varphi,ω,φ)V3​(φ,ω,φ), which includes the Toffoli gate as a special case. The derivation is fully analytical: we perform a sequence of unitary transformations, identify decoupled subspaces, and apply a rotating-wave approximation to obtain simple effective Hamiltonians. We derive explicit conditions on the coupling strengths and driving parameters, provide closed-form expressions for the time-evolution operators in each relevant subspace, and characterize the quality of the implementation using the operator fidelity. Numerical simulations show that the protocol achieves high fidelities over broad parameter ranges, demonstrating its robustness and suitability for quantum information processing in spin-chain platforms.