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-1/2 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.
No abstract
We give an analytic construction of a class of two-qubit gate pulse sequences that act on five of the six spin-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. We thereby analytically derive the Fong-Wandzura sequence that has so far escaped intuitive explanation. [7], and used to manipulate a variety of three-spin encoded qubits [8][9][10][11][12][13], including the so-called resonant exchange qubit [14][15][16] which maintains qubit encoding by keeping the exchange interaction "always on" within each qubit (see also [17]). Here we focus on the case of exchange-only quantum computation where the exchange interaction is kept completely off except when being pulsed, i.e. adiabatically switched on and off, between pairs of spins. It is then necessary to design pulse sequences that carry out quantum gates on encoded qubits without resulting in leakage out of the encoded qubit space [5,11,[18][19][20][21][22]. Control of theTo assess any quantum computation scheme one ultimately needs to know the minimal cost of carrying out quantum gates. For exchange-only quantum computation using pulse sequences, single-qubit gate sequences are theoretically understood but there is little true understanding regarding optimization of two-qubit gate sequences. The main difficulty comes from the constraint of no leakage combined with the large search space of unitary operators acting on the six spins encoding a pair of three-spin qubits. Not surprisingly, the shortest known pulse sequence for an entangling two-qubit gate due to Fong and Wandzura [20] has been found by a numerical search algorithm which offers little insight into its derivation. Furthermore, existing analytic derivations of less optimal sequences are lengthy and complicated [19,22].In this Letter we present an analytic construction of a class of pulse sequences that carry out two-qubit gates for exchange-only quantum computation. These sequences are built out of smaller sequences that act on only four , |a with a = 0, 1, and noncomputational state with total spin spins and satisfy a certain constraint. We show that when the most efficient of these smaller sequences is used the result is equivalent to the Fong-Wandzura sequence. Our guiding principle throughout is to avoid as much as possible complicated calculations and use only the most basic facts about quantum spin [23].Because we only consider the action of rotationally invariant operators, it is sufficient to describe quantum states of multiple spins using only total spin quantum numbers. Accordingly, we employ the notation used in [22] in which ea...
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