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arXiv:quant-ph/0410001 · Physical Review A

Optimal two-qubit quantum circuits using exchange interactions

Heng Fan, Vwani Roychowdhury, Thomas Szkopek

Published Nov 21, 2005• 47 citations
arXivHTMLPDF

Abstract

We give the optimal decomposition of a universal two-qubit circuit using Heisenberg exchange interactions and single qubit rotations. Tuning the strength and duration of the Heisenberg exchange allows one to implement ${(\mathrm{SWAP})}^{\ensuremath{\alpha}}$ gates. Our optimal circuit is constructed from only three ${(\mathrm{SWAP})}^{\ensuremath{\alpha}}$ gates and six single qubit gates. We show that three ${(\mathrm{SWAP})}^{\ensuremath{\alpha}}$ gates are not only sufficient, but necessary. Since six single-qubit gates are already known to be necessary, our implementation is optimal in gate count.

DOI
10.1103/physreva.72.052323

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We give the optimal decomposition of a universal two-qubit circuit using Heisenberg exchange interactions and single qubit rotations. Tuning the strength and duration of the Heisenberg exchange allows one to implement (SWAP)\ensuremathα{(\mathrm{SWAP})}^{\ensuremath{\alpha}}(SWAP)\ensuremathα gates. Our optimal circuit is constructed from only three (SWAP)\ensuremathα{(\mathrm{SWAP})}^{\ensuremath{\alpha}}(SWAP)\ensuremathα gates and six single qubit gates. We show that three (SWAP)\ensuremathα{(\mathrm{SWAP})}^{\ensuremath{\alpha}}(SWAP)\ensuremathα gates are not only sufficient, but necessary. Since six single-qubit gates are already known to be necessary, our implementation is optimal in gate count.
This paper addresses the question of how to implement a desired two-qubit gate U using a given tunable two-qubit entangling interaction . We present a general method which is based on the K 1 AK 2 decomposition of unitary matrices ∊ SU (4) to calculate the smallest number of two-qubit gates U int ( t ) (based on ) and single-qubit rotations, and the explicit sequence of these operations that are required to implement U . We illustrate our protocol by calculating the implementation of (1) the transformation from standard basis to Bell basis, (2) the CNOT-gate and (3) the quantum Fourier transform for two kinds of interaction—Heisenberg exchange interaction and quantum inductive coupling—and discuss the relevance of our results for solid-state qubits.
In this paper, we study the optimal simulation of three-qubit unitary by using two-qubit gates. First, we give a lower bound on the two-qubit gates cost of simulating a multi-qubit gate. Secondly, we completely characterize the two-qubit gate cost of simulating a three-qubit controlled controlled gate by generalizing our result on the cost of Toffoli gate. The function of controlled controlled gate is simply a three-qubit controlled unitary gate and can be intuitively explained as follows: the gate will output the states of the two control qubit directly, and apply the given one-qubit unitary uuu on the target qubit only if both the states of the control are ∣1⟩\ket{1}∣1⟩. Previously, it is only known that five two-qubit gates is sufficient for implementing such a gate [Sleator and Weinfurter, Phys. Rev. Lett. 74, 4087 (1995)]. Our result shows that if the determinant of uuu is 1, four two-qubit gates is achievable optimal. Otherwise, five is optimal. Thirdly, we show that five two-qubit gates are necessary and sufficient for implementing the Fredkin gate(the controlled swap gate), which settles the open problem introduced in [Smolin and DiVincenzo, Phys. Rev. A, 53, 2855 (1996)]. The Fredkin gate is one of the most important quantum logic gates because it is universal alone for classical reversible computation, and thus with little help, universal for quantum computation. Before our work, a five two-qubit gates decomposition of the Fredkin gate was already known, and numerical evidence of showing five is optimal is found.
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.
Quantum circuits currently constitute a dominant model for quantum computation [14]. Our work addresses the problem of constructing quantum circuits to implement an arbitrary given quantum computation, in the special case of two qubits. We pursue circuits without ancilla qubits and as small a number of elementary quantum gates [1, 9] as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2 CNOTs. To this end, we constructively prove a worstcase upper bound of 23 elementary gates, of which at most 4 (CNOTs) entail multi-qubit interactions. Our analysis shows that previously known synthesis algorithms, although more general, entail much larger quantum circuits than ours in the special case of two qubits. One such algorithm [4] has a worst case of 61 gates of which 18 may be CNOTs. Our techniques rely on the KAK decomposition from Lie theory as well as the polar and spectral (symmetric Shur) matrix decompositions from numerical analysis. They are related to the canonical decomposition of a two-qubit gate with respect to the "magic basis" of phase-shifted Bell states [11, 12]. We extend this decomposition in terms of elementary gates for quantum computation.
Quantum circuits currently constitute a dominant model for quantum computation [14]. Our work addresses the problem of constructing quantum circuits to implement an arbitrary given quantum computation, in the special case of two qubits. We pursue circuits without ancilla qubits and as small a number of elementary quantum gates [1, 9] as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2 CNOTs. To this end, we constructively prove a worstcase upper bound of 23 elementary gates, of which at most 4 (CNOTs) entail multi-qubit interactions. Our analysis shows that previously known synthesis algorithms, although more general, entail much larger quantum circuits than ours in the special case of two qubits. One such algorithm [4] has a worst case of 61 gates of which 18 may be CNOTs. Our techniques rely on the KAK decomposition from Lie theory as well as the polar and spectral (symmetric Shur) matrix decompositions from numerical analysis. They are related to the canonical decomposition of a two-qubit gate with respect to the "magic basis" of phase-shifted Bell states [11, 12]. We extend this decomposition in terms of elementary gates for quantum computation.