Effective nonlocal parity-dependent couplings in qubit chains

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Effective nonlocal parity-dependent couplings in qubit chains

Maximilian Nägele, Christian Schweizer, Federico Roy, and Stefan Filipp

Phys. Rev. Research 4, 033166 – Published 30 August 2022

Abstract

For the efficient implementation of quantum algorithms, practical ways to generate many-body entanglement are a basic requirement. Specifically, coupling multiple qubit pairs at once can be advantageous and may provide multiqubit operations useful in the construction of hardware-tailored algorithms. Here we extend the theory of fractional state transfer and harness the simultaneous coupling of qubits on a chain to engineer a set of nonlocal parity-dependent quantum operations suitable for a wide range of applications. The resulting effective long-range couplings directly implement a parametrizable Trotter-step for Jordan-Wigner fermions, and they can be used for simulations of quantum dynamics, efficient state generation in variational quantum eigensolvers, parity measurements for error-correction schemes, and the generation of efficient multiqubit gates. Moreover, we present numerical simulations of the gate operation in a superconducting quantum circuit architecture, which show a high gate fidelity for realistic experimental parameters.

Received 3 May 2022
Revised 21 July 2022
Accepted 26 July 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.033166

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum algorithms Quantum engineering Quantum entanglement Quantum error correction Quantum gates Quantum simulation

Quantum Information, Science & Technology

Authors & Affiliations

Maximilian Nägele ^1,2, Christian Schweizer ^1,2,3, Federico Roy ^2,4, and Stefan Filipp ^2,3,5

¹Fakultät für Physik, Ludwig-Maximilians-Universität München, Schellingstraße 4, D-80799 München, Germany
²Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany
³Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany
⁴Theoretical Physics, Saarland University, 66123 Saarbrücken, Germany
⁵Physik-Department, Technische Universität München, 85748 Garching, Germany

Article Text

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Issue

Vol. 4, Iss. 3 — August - October 2022

Subject Areas

Quantum Information

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Images

Figure 1
Qubit chain and effective parity-dependent couplings. (a) Chain of qubits (circles) with frequencies $Δ_{n}$ and direct couplings $J_{n}$ , as described by $H_{N}$ in Eq. (1). Dark red lines indicate effective nonlocal interactions that stroboscopically arise for specific parameter choices of $Δ_{n}$ and $J_{n}$ . The effective interaction results in a rotation in the subspaces spanned by $| n 〉$ and $| N + 1 - n 〉$ , where $n$ denotes the location of the excitation. The orientation of the rotation vector depends on the parity of the qubits between each pair, $\otimes_{k = n + 1}^{N - n} Z_{k}$ . (b) Illustration of a chain with length $N = 3$ . A single excitation is prepared at site 1 and partially transferred to site 3 with effective interaction $σ_{1}^{+} σ_{3}^{-} + H.c.$ (left chain), which rotates the state by an angle $θ$ on the Bloch-sphere spanned by the states $| 1 〉$ and $| 3 〉$ (red arrow). If an additional excitation is prepared at site 2 (right chain), the effective interaction changes sign, so that the state is rotated by an angle $- θ$ on the Bloch-sphere (blue arrow).
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Figure 2
Simulated occupation dynamics during two consecutive fractional state transfers (FST) on an $N = 15$ qubit chain. Hamiltonian parameters are chosen to achieve a transfer angle $θ = π / 2$ . (a) Evolution of an excitation prepared at site 1. After a first FST at time $τ$ (dotted line), the chain is in a superposition of the excitation being at either of its ends. In the Bloch-sphere spanned by the states $| 1 〉$ and $| 15 〉$ , this corresponds to a rotation by an angle $π / 2$ (dark red arrow). At time $2 τ$ after a second FST, the excitation refocuses at site 15, resulting in a $2 θ = π$ rotation on the Bloch-sphere (gray arrow). (b) Evolution of a state with excitations prepared at sites 1 and 8. In this case, the first FST rotates the state $| 1 〉$ on the Bloch-sphere by a negative angle $- π / 2$ (dark blue arrow), due to the odd parity of excitations in the middle of the chain. In contrast, the center excitation refocuses at its original location and is then removed by an instantaneous $π$ -flip gate $X_{8}^{π}$ at site 8 (blue rectangle), which changes the parity. Then, with a second FST the dynamics are reverted and the excitation refocuses at site 1.
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Figure 3
Comparison between fractional state transfer (FST) and equivalent decomposition. (a) An even chain folded in the middle forms a ladder, where FST introduces effective interactions along its rungs corresponding to the different terms of $G_{N}$ (red lines). (b) Decomposition of FST into two-qubit gates for even chains. Steps one (red and dark blue arrows) and two (light blue arrows) are repeated $N / 2$ times. Blue arrows symbolize FSWAP gates, while the red arrow is an iSWAP $(θ)$ gate. Odd chains are discussed in Appendix pp3. (c) Relative speed gain of FST with respect to the two-qubit gate decomposition of $exp (- i θ G_{N} / 2)$ as a function of the chain length $N$ for both odd (red) and even (blue) $N$ . The speed of FST is assumed to be limited by the maximum coupling in the chain as given in Eq. (A1). For $N > 4$ and perfect state transfer, i.e., $θ = π$ , FST is at least a factor of 2 (upper solid line) faster. For $N > 4$ and transfer angles approaching zero, i.e., $θ \to 0$ , the same speedup is still present for $N$ odd but reduces to $\sqrt{3}$ (lower solid line) for $N$ even. $N = 3$ and 4 are special cases since some steps of the decomposition contain only iSWAP $(θ)$ gates, which are faster for smaller $θ$ . The point for $N = 3$ and $θ \to 0$ is not shown since $t_{FST} \to 0$ in this case.
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Figure 4
Simulation of the three-qubit fractional-state-transfer gate for superconducting qubits. (a) Three fixed-frequency transmons ( $Q_{1}, Q_{2}, Q_{3}$ ) are coupled via two tunable couplers ( $C_{1}, C_{2}$ ). By periodically modulating the coupler frequencies through flux pulses ( $Φ_{c 1}, Φ_{c 2}$ ), effective interactions between the qubits arise ( $J_{eff}$ ). (b) Infidelity and leakage (population losses out of the computational subspace averaged over the computational states) of the optimized gate at various angles. (c) The total gate time for different $θ$ is set as theoretically predicted (solid red). The gate time is significantly shorter than the decomposition into two-qubit gates (dashed blue). (d) Detuning of both drives (markers lie exactly over each other) and theoretical prediction (dashed line). The detuning is slightly shifted from the prediction because of ac-stark shifts.
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Figure 5
Hamiltonian parameters $J_{n}$ and $Δ_{n}$ as a function of the transfer angle $θ$ for chain length $N = 9$ in (a) and (b) and $N = 10$ in (c). (a) The magnitude of the required detunings decreases linearly with $θ$ if $N$ is odd. Perfect state transfer requires no detunings. For $N$ even, no detunings are needed. (b) All coupling strengths increase with $θ$ if $N$ is odd. (c) For $N$ even, only the center coupling increases with $θ$ while all other couplings decrease.
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Figure 6
Building blocks of the decomposition of the FST gate into two qubit gates. The chain is folded in half at the middle forming a ladder. Yellow arrows symbolize FSWAP gates. Red dotted arrows symbolize an $iSWAP (- θ)$ gate. (a) For $N$ even, the FST gate is decomposed by $N / 2$ applications of $U_{1}^{e} U_{2}^{e}$ . (b) For $N$ odd, the FST gate is decomposed by a single application of $U_{start}^{o}$ followed by $(N - 1) / 2$ applications of $U_{1}^{o} U_{2}^{o}$ and a final application of $U_{final}^{o}$ .
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