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arXiv:1508.07524 · Physical Review A

Simple derivation of the Fong-Wandzura pulse sequence

Daniel Zeuch, N. E. Bonesteel

Published Jan 25, 2016• 14 citations
arXivHTMLPDF

Abstract

We give an analytic construction of a class of two-qubit gate pulse sequences that act on five of the six spin-$\frac{1}{2}$ particles used to encode a pair of exchange-only three-spin qubits. Within this class, the problem of gate construction reduces to that of finding a smaller sequence that acts on four spins and is subject to a simple constraint. The optimal sequence satisfying this constraint yields a two-qubit gate sequence equivalent to that found numerically by Fong and Wandzura. Our construction is sufficiently simple that it can be carried out entirely with pen, paper, and knowledge of a few basic facts about quantum spin.

DOI
10.1103/physreva.93.010303

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We give an analytic construction of a class of two-qubit gate pulse sequences that act on five of the six spin-12\frac{1}{2}21​ particles used to encode a pair of exchange-only three-spin qubits. Within this class, the problem of gate construction reduces to that of finding a smaller sequence that acts on four spins and is subject to a simple constraint. The optimal sequence satisfying this constraint yields a two-qubit gate sequence equivalent to that found numerically by Fong and Wandzura. Our construction is sufficiently simple that it can be carried out entirely with pen, paper, and knowledge of a few basic facts about quantum spin.
We design efficient controlled-rotation gates with arbitrary angle acting on three-spin encoded qubits for exchange-only quantum computation. Two pulse sequence constructions are given. The first is motivated by an analytic derivation of the efficient Fong-Wandzura sequence for an exact cnot gate. This derivation, briefly reviewed here, is based on elevating short sequences of swap pulses to an entangling two-qubit gate. To go beyond cnot, we apply a similar elevation to a modified short sequence consisting of swaps and one pulse of arbitrary duration. This results in two-qubit sequences that carry out controlled-rotation gates of arbitrary angle. The second construction streamlines a class of arbitrary cphase gates established earlier. Both constructions are based on building two-qubit sequences out of subsequences with special properties that render each step of the construction analytically tractable.
We present pulse sequences for two-qubit gates acting on encoded qubits for exchange-only quantum computation. Previous work finding such sequences has always required numerical methods due to the large search space of unitary operators acting on the space of the encoded qubits. By contrast, our construction can be understood entirely in terms of three-dimensional rotations of effective spin-12\frac{1}{2}21​ pseudospins which allows us to use geometric intuition to determine the required sequence of operations analytically. The price we pay for this simplification is that, at 39 pulses, our sequences are significantly longer than the best numerically obtained sequences.
Exchange-only (EO) spin qubits in quantum dots offer an expansive design landscape for engineering scalable device layouts. The study of two-EO-qubit operations, which involve six electrons in six quantum dots, has so far been limited to a small number of possible configurations, and previous works lack analyses of design considerations and implications for quantum error correction. Using a simple and fast optimization method, we generate complete pulse sequences for , , i , leakage-controlled , and leakage-controlled two-qubit gates on 450 unique planar six-dot topologies and analyze differences in sequence length (up to 43% reduction) across topology classes. In addition, we show that relaxing constraints on postoperation spin locations can yield further reductions in sequence length; conversely, constraining these locations in a particular way generates a operation with minimal additional cost over a standard . We integrate this pulse library into the Intel quantum stack and experimentally verify pulse sequences on a Tunnel Falls chip for different operations in a linear-connectivity device to confirm that they work as expected. Finally, we explore architectural implications of these results for quantum error correction. Our work guides hardware and software design choices for future implementations of scalable quantum-dot architectures.
The benefit of exchange-only qubits compared to other spin-qubit types is the universal control using only voltage-controlled exchange interactions between neighboring spins. As a compromise, qubit operations have to be constructed from nonorthogonal rotation axes of the Bloch sphere and result in rather long pulsing sequences. This paper aims to develop a faster implementation of single-qubit gates using simultaneous exchange pulses and manifests their potential for the construction of two-qubit gates. We introduce pulse sequences in which single-qubit gates could be executed faster and show that subsequences on three spins in two-qubit gates could be implemented in fewer steps. Our findings can particularly speed up gate sequences for realistic idle times between sequential pulses and we show that this advantage increases with more interconnectivity of the quantum dots. We further demonstrate how a phase operation can introduce a relative phase between the computational and some of the leakage states, which can be advantageous for the construction of two-qubit gates. In addition to our theoretical analysis, we experimentally demonstrate and characterize a simultaneous exchange implementation of X rotations in a Si Ge quantum dot device and compare to the state of the art with sequential exchange pulses.
Pairwise exchange couplings have long been the standard mechanism for entangling spin qubits in semiconductor systems. However, implementing quantum circuits based on pairwise exchange gates often requires a lengthy sequence of elementary gate operations. In this work, we present an alternative approach: multi-qubit entangling gate operations that simultaneously drive the exchange couplings between multiple pairs of spin qubits. We explore three spin qubit systems in linear or triangular configurations. We derive analytical expressions for these multi-exchange entangling operations and demonstrate how to use the resulting three-qubit gates to construct quantum circuits capable of generating standard entangled states such as GHZ and W states, and the Toffoli gate, by optimizing control parameters. Our results show that this multi-qubit strategy significantly reduces the number of required operations, offering a pathway to more efficient, shallower, and more coherent circuits for spin-qubit processors.
Efficient operation sequences to couple and interchange quantum information between quantum dot spin qubits of different kinds are derived using exchange interactions. In the qubit encoding of a single-spin qubit, a singlet-triplet qubit, and an exchange-only (triple-dot) qubit, some of the single-qubit interactions remain on during the entangling operation; this greatly simplifies the operation sequences that construct entangling operations. In the ideal setup, the gate operations use the intraqubit exchange interactions only once, and entangling operations with gate times similar to typical single-qubit operations are constructed. The limitations of the entangling sequences are discussed, and it is shown how quantum information can be converted between different kinds of quantum dot spin qubits. These gate sequences are useful for large-scale quantum computation because they show that different kinds of coded spin qubits can be combined easily, permitting the favorable physical properties of each to be employed.