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arXiv:2307.11900

Efficient, direct compilation of SU(N) operations into SNAP & Displacement gates

Joshua Job

Published Jul 21, 2023• 1 citation
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Abstract

We present a function which connects the parameter of a previously published short sequence of selective number-dependent arbitrary phase (SNAP) and displacement gates acting on a qudit encoded into the Fock states of a superconducting cavity, $V_k(α)=D(α)R_π(k)D(-2α)R_π(k)D(α)$ to the angle of the Givens rotation $G(θ)$ on levels $|k\rangle,|k+1\rangle$ that sequence approximates, namely $α=Φ(θ) = \fracθ{4\sqrt{k+1}}$. Previous publications left the determination of an appropriate $α$ to numerical optimization at compile time. The map $Φ$ gives us the ability to compile directly any $d$-dimensional unitary into a sequence of SNAP and displacement gates in $O(d^3)$ complex floating point operations with low constant prefactor, avoiding the need for numerical optimization. Numerical studies demonstrate that the infidelity of the generated gate sequence $V_k$ per Givens rotation $G$ scales as approximately $O(θ^6)$. We find numerically that the error on compiled circuits can be made arbitrarily small by breaking each rotation into $m$ $θ/m$ rotations, with the full $d\times d$ unitary infidelity scaling as approximately $O(m^{-4})$. This represents a significant reduction in the computational effort to compile qudit unitaries either to SNAP and displacement gates or to generate them via direct low-level pulse optimization via optimal control.

DOI
10.48550/arxiv.2307.11900

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We present a function which connects the parameter of a previously published short sequence of selective number-dependent arbitrary phase (SNAP) and displacement gates acting on a qudit encoded into the Fock states of a superconducting cavity, Vk(α)=D(α)Rπ(k)D(−2α)Rπ(k)D(α)V_k(α)=D(α)R_π(k)D(-2α)R_π(k)D(α)Vk​(α)=D(α)Rπ​(k)D(−2α)Rπ​(k)D(α) to the angle of the Givens rotation G(θ)G(θ)G(θ) on levels ∣k⟩,∣k+1⟩|k\rangle,|k+1\rangle∣k⟩,∣k+1⟩ that sequence approximates, namely α=Φ(θ)=θ4k+1α=Φ(θ) = \fracθ{4\sqrt{k+1}}α=Φ(θ)=4k+1​θ​. Previous publications left the determination of an appropriate ααα to numerical optimization at compile time. The map ΦΦΦ gives us the ability to compile directly any ddd-dimensional unitary into a sequence of SNAP and displacement gates in O(d3)O(d^3)O(d3) complex floating point operations with low constant prefactor, avoiding the need for numerical optimization. Numerical studies demonstrate that the infidelity of the generated gate sequence VkV_kVk​ per Givens rotation GGG scales as approximately O(θ6)O(θ^6)O(θ6). We find numerically that the error on compiled circuits can be made arbitrarily small by breaking each rotation into mmm θ/mθ/mθ/m rotations, with the full d×dd\times dd×d unitary infidelity scaling as approximately O(m−4)O(m^{-4})O(m−4). This represents a significant reduction in the computational effort to compile qudit unitaries either to SNAP and displacement gates or to generate them via direct low-level pulse optimization via optimal control.
Microwave cavities coupled to superconducting qubits have been demonstrated to be a promising platform for quantum information processing. A major challenge in this setup is to realize universal control over the cavity. A promising approach are selective number-dependent arbitrary phase (SNAP) gates combined with cavity displacements. It has been proven that this is a universal gate set, but a central question remained open so far: how can a given target operation be realized efficiently with a sequence of these operations. In this work, we present a practical scheme to address this problem. It involves a hierarchical strategy to insert new gates into a sequence, followed by a co-optimization of the control parameters, which generates short high-fidelity sequences. For a broad range of experimentally relevant applications, we find that they can be implemented with 3 to 4 SNAP gates, compared to up to 50 with previously known techniques.
Advanced simulations and calculations on quantum computers require high-fidelity implementations of quantum operations. The universal gateset approach builds complex unitaries from a small set of primitive gates, often resulting in a long gate sequence, which is typically a leading factor in the total accumulated error. Compiling a complex unitary for processors with higher-dimensional logical elements, such as qutrits, exacerbates the accumulated error per unitary, since an even longer gate sequence is required. Optimal control methods promise time- and resource-efficient compact gate sequences and, therefore, higher fidelity. These methods generate pulses that can directly implement any complex unitary on a quantum device. In this work, we demonstrate that any arbitrary qubit and qutrit gate can be realized with high fidelity, which can significantly reduce the length of a gate sequence. We generate and test pulses for a large set of randomly selected arbitrary unitaries on several quantum processing units (QPUs): the Lawrence Livermore National Laboratory Quantum Device and Integration Testbed's (QuDIT's) standard QPU and three of Rigetti's QPUs: Ankaa-2, Ankaa-9Q-1, and Aspen-M-3. On the QuDIT platform's standard QPU, the average fidelity of random qutrit gates is $97.9\ifmmode\pm\else\textpm\fi{}0.5\mathrm{%}$ measured with conventional QPT and $98.8\ifmmode\pm\else\textpm\fi{}0.6\mathrm{%}$ from QPT with gate folding. Rigetti's Ankaa-2 achieves random qubit gates with an average fidelity of $98.4\ifmmode\pm\else\textpm\fi{}0.5\mathrm{%}$ (conventional QPT) and $99.7\ifmmode\pm\else\textpm\fi{}0.1\mathrm{%}$ (QPT with gate folding). On Ankaa-9Q-1 and Aspen-M-3, the average fidelities with conventional qubit QPT measurements were higher than 99% (see Appendix). We show that optimal control gates are robust to drift for at least 3 h and that the same calibration parameters can be used for all implemented gates. Our work promises that the calibration overheads for optimal control gates can be made small enough to enable efficient quantum circuits based on this technique.
In this work, we investigate the effect of decomposition basis on primitive qudit gates on superconducting radio-frequency cavity-based quantum computers with applications to lattice gauge theory. Three approaches are tested: SNAP & Displacement gates, ECD & single-qubit rotations R(θ,ϕ)R(θ,ϕ)R(θ,ϕ), and optimal pulse control. For all three decompositions, implementing the necessary sequence of rotations concurrently rather then sequentially can reduce the primitive gate run time. The number of blocks required for the faster ECD & Rp(θ)R_p(θ)Rp​(θ) is found to scale O(d2)\mathcal{O}(d^2)O(d2), while slower SNAP & Displacement set scales at worst O(d)\mathcal{O}(d)O(d). For qudits with d<10d<10d<10, the resulting gate times for the decompositions is similar, but strongly-dependent on experimental design choices. Optimal control can outperforms both decompositions for small ddd by a factor of 2-12 at the cost of higher classical resources. Lastly, we find that SNAP & Displacement are slightly more robust to a simplified noise model.
Preparing quantum states in bosonic qudits with SNAP--Displacement sequences requires both optimizing the gate parameters and deciding which Fock levels should be controlled by each SNAP gate. A fully parameterized ansatz assigns an independent phase to every Fock level in every gate, leading to a number of phases that grows steeply. This constitutes a problem for short to medium term systems as controlling many phases requires complex control pulses which are subject to increased control errors. Previous work reduced this cost by restricting the trainable phases to predefined sparse patterns, such as a diagonal or a grid, which however may not be optimal for a specific target state. In this work, we address this limitation by introducing a method that learns which phases are needed together with their values, using a differentiable mask. A sparsity penalty encourages the removal of unnecessary phases, while temperature annealing gradually turns the masks into binary decisions. We compare the proposed method with fully parameterized and predefined sparse ansatzes on Fourier and Haar-random target states. The evaluation considers both the minimum number of phases required to reach different infidelity thresholds and the overall trade-off between fidelity and phases required, measured through Pareto frontiers and hypervolume. The results show that learning the location of the phases together with their values provides competitive or improved fidelity--resource trade-offs, although the advantage depends on the target state and qudit dimension.
Abstract In this study, we explore the universality of Selective Number-dependent Arbitrary Phase (SNAP) and Displacement gates for quantum control in qudit-based systems. However, optimizing the parameters of these gates poses a challenging task. Our main focus is to investigate the sensitivity of training any of the SNAP parameters in the SNAP-Displacement protocol. We analyze conditions that could potentially lead to the Barren Plateau problem in a qudit system and draw comparisons with multi-qubit systems. The parameterized ansatz we consider consists of blocks, where each block is composed of hardware operations, namely SNAP and Displacement gates [Fösel et al 2020 Efficient cavity control with snap gates arXiv:2004.14256]. Applying Variational Quantum algorithm (VQA) with observable and gate cost functions, we utilize techniques similar to those in [McClean et al 2018 Barren plateaus in quantum neural network training landscapes Nat. Commun. 9 1–6] and [Cerezo et al 2021 Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nat. Commun. 12 1–12] along with the concept of t − design. Through this analysis, we make the following key observations: (a) The trainability of a SNAP-parameter does not exhibit a preference for any particular direction within our cost function landscape, (b) By leveraging the first and second moments properties of Haar measures, we establish new lemmas concerning the expectation of certain polynomial functions, and (c) utilizing these new lemmas, we identify a general condition that indicates an expected trainability advantage in a qudit system when compared to multi-qubit systems.
The large available Hilbert space and high coherence of cavity resonators make these systems an interesting resource for storing encoded quantum bits. To perform a quantum gate on this encoded information, however, complex nonlinear operations must be applied to the many levels of the oscillator simultaneously. In this work, we introduce the selective number-dependent arbitrary phase (snap) gate, which imparts a different phase to each Fock-state component using an off-resonantly coupled qubit. We show that the snap gate allows control over the quantum phases by correcting the unwanted phase evolution due to the Kerr effect. Furthermore, by combining the snap gate with oscillator displacements, we create a one-photon Fock state with high fidelity. Using just these two controls, one can construct arbitrary unitary operations, offering a scalable route to performing logical manipulations on oscillator-encoded qubits.