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arXiv:quant-ph/0310094 · Physical Review Letters

One-Spin Quantum Logic Gates from Exchange Interactions and a Global Magnetic Field

Lian-Ao Wu, Daniel A. Lidar, Mark G. Friesen

Published Jul 15, 2004• 37 citations
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Abstract

It has been widely assumed that one-qubit gates in spin-based quantum computers suffer from severe technical difficulties. We show that one-qubit gates can, in fact, be generated using only modest and presently feasible technological requirements. Our solution uses only global magnetic fields and controllable Heisenberg exchange interactions, thus circumventing the need for single-spin addressing.

DOI
10.1103/physrevlett.93.030501

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It has been widely assumed that one-qubit gates in spin-based quantum computers suffer from severe technical difficulties. We show that one-qubit gates can, in fact, be generated using only modest and presently feasible technological requirements. Our solution uses only global magnetic fields and controllable Heisenberg exchange interactions, thus circumventing the need for single-spin addressing.
The construction of a solid-state quantum computer remains a coveted, yet elusive goal. Among the most promising scalable candidates are the spin-based approaches, where qubits are represented by electron spins in quantum dots, or by donor-atom nuclear or electron spins, or by combinations thereof. It has been widely assumed that one-qubit gates in these systems, which are based on single-spin addressing, suffer from severe technical difficulties, including slow speed of operation, heating, and device and materials complexity, such as g-factor engineering. For this reason a variety of approaches have been suggested to circumvent one-qubit gates, including encoding the state of a qubit into two or more spins. These approaches inevitably introduce substantial overhead. Here we show that one-qubit gates can in fact be generated using only modest and presently feasible technological requirements. Our solution uses only global magnetic fields and controllable Heisenberg exchange interactions, thus circumventing the need for single-spin addressing. We explicitly show how to build up one-qubit gates from these components using simple pulse sequences. This should substantially simplify the implemention of spin-based solid state quantum computers.
We propose a method that implements a universal set of one- and two-quantum-bit gates for quantum computation in a system of coupled electron pairs with constant non-diagonal exchange interaction. In our proposal, suppression of the exchange interaction is performed by the continual repetition of single-spin rotations. A small g-factor difference between the electrons allows for addressing qubits and avoiding strong magnetic field pulses. Numerical experiments were performed to show that, to implement the one- and two-qubit operations, it is sufficient to change the strength of the magnetic field by a few Gauss. This introduces one and then the other electron in a resonance. To determine the evolution of the two-qubit system, we use the algorithms of optimal control theory.
We study the low energy states of one-dimensional spin chain clusters consisting of three spin-1/2 Heisenberg particles. When applying an external magnetic field on the spin chain cluster, the ground state doublet will generate Zeeman splitting. As a two-level quantum system, the split ground state doublet can be defined as a new 'spin cluster qubit'. The cluster quibit is shown to be more robust with a smaller decoherence rate. We realize universal single-qubit gates with the algebraic dynamical method, by applying a time-dependent magnetic field on the spin cluster. We also investigate a two-qubit system made up of two such Heisenberg spin chain clusters which interact through a tunable Heisenberg coupling. In the two-qubit product basis, the controlled quantum phase gates can be generated by applying a magnetic field along the z-axis on the system of the two clusters. One can realize the controlled-NOT gate through a three-step time evolution by adjusting the corresponding physical parameters. Our scheme involves two kinds of dynamically distinct operations, which is much simpler and more feasible.
We study the low energy states of finite spin chains with isotropic (Heisenberg) and anisotropic (XY and Ising-like) antiferromagnetic exchange interaction with uniform and nonuniform coupling constants. We show that for an odd number of sites a spin cluster qubit can be defined in terms of the ground state doublet. This qubit is remarkably insensitive to the placement and coupling anisotropy of spins within the cluster. One- and two-qubit quantum gates can be generated by magnetic fields and intercluster exchange, and leakage during quantum gate operation is small. Spin cluster qubits inherit the long decoherence times and short gate operation times of single spins. Control of single spins is hence not necessary for the realization of universal quantum gates.
We show that a wide range of spin clusters with antiferromagnetic intracluster exchange interaction allows one to define a qubit. For these spin cluster qubits, initialization, quantum gate operation, and readout are possible using the same techniques as for single spins. Quantum gate operation for the spin cluster qubit does not require control over the intracluster exchange interaction. Electric and magnetic fields necessary to effect quantum gates need only be controlled on the length scale of the spin cluster rather than the scale for a single spin. Here, we calculate the energy gap separating the logical qubit states from the next excited state and the matrix elements which determine quantum gate operation times. We discuss spin cluster qubits formed by one- and two-dimensional arrays of s=1/2s=1/2s=1/2 spins as well as clusters formed by spins s>1/2.s>1/2.s>1/2. We illustrate the advantages of spin cluster qubits for various suggested implementations of spin qubits and analyze the scaling of decoherence time with spin cluster size.
We show how to construct a universal set of quantum logic gates using control over exchange interactions and single- and two-spin measurements only. Single-spin unitary operations are teleported between neighboring spins instead of being executed directly, thus potentially eliminating a major difficulty in the construction of several of the most promising proposals for solid-state quantum computation, such as spin-coupled quantum dots, donor-atom nuclear spins in silicon, and electrons on helium. Contrary to previous proposals dealing with this difficulty, our scheme requires no encoding redundancy. We also discuss an application to superconducting phase qubits.