Simple operation sequences to couple and interchange quantum information between spin qubits of different kinds

Sebastian Mehl and David P. DiVincenzo

Phys. Rev. B 92, 115448 – Published 30 September 2015

Abstract

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.

Received 13 July 2015
Revised 7 September 2015

DOI:https://doi.org/10.1103/PhysRevB.92.115448

Authors & Affiliations

Sebastian Mehl^1,2,* and David P. DiVincenzo^1,2

¹JARA-Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany
²Peter Grünberg Institute (PGI-2), Forschungszentrum Jülich, D-52425 Jülich, Germany

^*s.mehl@fz-juelich.de

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Vol. 92, Iss. 11 — 15 September 2015

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Figure 1
Entangling operation between a single-spin qubit and a STQ. (a) ${QD}_{1}$ defines a single-spin qubit with the qubit levels ${|0^{L}〉, |1^{L}〉}$ ; ${QD}_{2}$ and ${QD}_{3}$ define a STQ with the qubit levels ${|0^{R}〉, |1^{R}〉}$ . A weak tunnel coupling between ${QD}_{1}$ and ${QD}_{2}$ couples the single-spin qubit and the STQ. (b) Sequence to create a CPHASE between a single-spin qubit (coded on ${QD}_{1})$ and a STQ (coded on ${QD}_{2}$ and ${QD}_{3})$ . $Z_{ϕ}^{L}$ and $Z_{ϕ}^{R}$ are the phase gates of the qubits L and R. $U_{ϕ, ψ}^{A}$ is defined in Eq. (2).
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Figure 2
Entangling operation between a single-spin qubit and an exchange-only qubit. (a) ${QD}_{1}$ defines a single-spin qubit with the qubit levels ${|0^{L}〉, |1^{L}〉}$ ; ${QD}_{2} − {QD}_{4}$ define an exchange-only qubit with the qubit levels ${|0^{R}〉, |1^{R}〉}$ . A weak tunnel coupling between ${QD}_{1}$ and ${QD}_{2}$ couples the single-spin qubit and the exchange-only qubit. (b) Sequence to create a CPHASE between a single-spin qubit (coded on ${QD}_{1})$ and an exchange-only qubit (coded on ${QD}_{2} − {QD}_{4})$ . $Z_{ϕ}^{L}$ and $Z_{ϕ}^{R}$ are the phase gates of the qubits L and R. $U_{ϕ}^{B}$ is defined in Eq. (8).
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Figure 3
Entangling operations between a STQ and an exchange-only qubit. ${QD}_{1}$ and ${QD}_{2}$ define a STQ with the qubit levels ${|0^{L}〉, |1^{L}〉}$ ; ${QD}_{3} − {QD}_{5}$ define an exchange-only qubit with the qubit levels ${|0^{R}〉, |1^{R}〉}$ . A weak tunnel coupling between ${QD}_{2}$ and ${QD}_{3}$ couples the STQ and the exchange-only qubit. (b) and (c) Sequences to create a CPHASE between a STQ (coded on ${QD}_{1}$ and ${QD}_{2})$ and an exchange-only qubit (coded on ${QD}_{3} − {QD}_{5})$ . $Z_{ϕ}^{L}$ and $Z_{ϕ}^{R}$ are the phase gates of the qubits L and R. $U_{ϕ, ψ}^{C_{1}}$ and $U_{ϕ, ψ}^{C_{2}}$ are defined in Eqs. (12) and (16). The CPHASE gate is abbreviated as CZ, and $H^{L}$ is the Hadamard gate for qubit L.
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Figure 4
Gate errors for the operation sequences of Eqs. (9), (13), and (17). Only the operations $U_{3 / 4}^{B}, U_{3 / (2 \sqrt{2}), 1 / \sqrt{2}}^{C_{1}}$ , and $U_{3 / \sqrt{2}, 1 / (2 \sqrt{2})}^{C_{2}}$ are analyzed. The gate errors are characterized by the deviation of the entanglement fidelity $F = tr (ρ^{RS} U_{ideal}^{- 1} U_{real} ρ^{RS} U_{real}^{- 1} U_{ideal})$ from 1. $ρ^{RS} = |RS〉 〈RS|$ is the maximally entangled state of two identical subspaces R and S, e.g., $|RS〉 \propto |0000〉 + |0110〉 + |1001〉 + |1111〉$ , and the time evolutions $U_{ideal}$ and $U_{real}$ act only on S while R remains unchanged. (a) For $U_{3 / 4}^{B}$ (blue curve) and $U_{3 / (2 \sqrt{2}), 1 / \sqrt{2}}^{C_{1}}$ (red curve), the exchange interaction of the exchange-only qubit $J$ should be by more than one order of magnitude larger than $J_{12}$ to reduce the gate error below 1%. (b) For $U_{3 / \sqrt{2}, 1 / (2 \sqrt{2})}^{C_{2}}, J_{12}$ , and $J$ should be large. The gate errors increase for $J_{12} = 3 J / 2, J_{12} = J$ , and $J_{12} = J / 2$ (dashed lines) because of degeneracies in the level spectrum.
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Figure 5
Gate operations to interchange qubits using CPHASE gates. (a) The unconditioned SWAP operation requires three CPHASE gates together with Hadamard gates (H). (b) A simpler SWAP sequence can be realized if one of the qubits is initialized to a fixed state, e.g., $|0〉$ . Then the SWAP operation with an arbitrary state $|ψ〉$ requires only two CPHASE gates.
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