Demonstration of Algorithmic Quantum Speedup

Bibek Pokharel and Daniel A. Lidar

Phys. Rev. Lett. 130, 210602 – Published 26 May 2023

Abstract

Despite the development of increasingly capable quantum computers, an experimental demonstration of a provable algorithmic quantum speedup employing today’s non-fault-tolerant devices has remained elusive. Here, we unequivocally demonstrate such a speedup within the oracular model, quantified in terms of the scaling with the problem size of the time-to-solution metric. We implement the single-shot Bernstein-Vazirani algorithm, which solves the problem of identifying a hidden bitstring that changes after every oracle query, using two different 27-qubit IBM Quantum superconducting processors. The speedup is observed on only one of the two processors when the quantum computation is protected by dynamical decoupling but not without it. The quantum speedup reported here does not rely on any additional assumptions or complexity-theoretic conjectures and solves a bona fide computational problem in the setting of a game with an oracle and a verifier.

Received 19 November 2022
Accepted 20 April 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.210602

Physics Subject Headings (PhySH)

Quantum algorithms

Quantum Information, Science & Technology

Authors & Affiliations

Bibek Pokharel^1,* and Daniel A. Lidar ^2,†

¹Department of Physics & Astronomy and Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA
²Departments of Electrical & Computer Engineering, Chemistry, and Physics & Astronomy, and Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA

^*pokharel@usc.edu
^†lidar@usc.edu

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Issue

Vol. 130, Iss. 21 — 26 May 2023

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Images

Figure 1
Circuit for the BV algorithm, including dynamical decoupling (DD) pulses. The oracle shown encodes the unknown bitstring $b = 111 000$ for the ssBV-6 problem. A controlled-NOT (CNOT, or CX) or identity gate is performed from qubit $i$ to the ancilla qubit if $b_{i} = 1$ or 0, respectively. Note that the quantum and classical oracles are identical in the ssBV- $n$ problem, and so both take time $t_{r} \propto | b |$ to run, where $| b |$ is the Hamming weight of $b$ . Each BV- $n$ circuit requires $n + 1$ qubits. A Hadamard gate ( $H$ ) is applied to each qubit before and after the oracle, and each qubit is measured in the computational basis for a total circuit $depth \geq | b | + 3$ (with equality only for fully connected architectures). DD pulses ( $P_{i}$ ) are turned on during idle times. Pulse placement is schematic but illustrates the principles we used in practice: (i) DD fills all available idle times; (ii) pulse intervals are varied depending on the available idle time per qubit. The actual timeline is shown in units of $d t = 2 / 9 ns$ —the inverse sampling rate of the backend’s arbitrary waveform generators.
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Figure 2
Full output distribution for BV-6 from Cairo. Oracles $f_{b}$ are numbered from 0 to 63, corresponding to $b \in {0^{6}, \dots, 1^{6}}$ , sorted by increasing Hamming weight. Ideally, the output state for oracle $f_{b}$ (vertical axis) is $b$ , but in reality, other bitstrings (horizontal axis) are observed as well. Green dots on the diagonal correspond to $p_{s} > 1 / 2$ , where $p_{s}$ is the empirical frequency (success probability) with which $b$ was output for oracle $f_{b}$ . Success probabilities are reported with $5 σ$ confidence intervals.
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Figure 3
Time to solution (TTS) as a function of problem size or number of data qubits $n$ . We report $TTS (n) = (1 / 2^{n}) \sum_{b} TTS (n, b)$ , where $TTS (n, b)$ is given by Eq. (1), with $p_{d} = 0.99$ and $t_{r} (n)$ replaced by $t_{r} (n, b)$ , since each oracle (labeled by $b \in {0, 1}^{n}$ ) takes a different time to run. Results for Montreal and Cairo are shown by the orange and blue symbols, respectively, and filled (empty) symbols represent results with (without) DD; dotted lines are guides to the eye. The asymptotic classical scaling ${TTS}_{C} (n) \sim 2^{n}$ is shown as white grid lines, and the hypothetical, ideal quantum scaling ${TTS}_{Q} (n) \propto n$ of each QC is indicated by the dashed lines (for QC-specific parameter values see [30]). The solid lines give the worst-case scaling fit for each curve, whose slopes $λ$ are reported in the bottom legend, with uncertainties representing 95% confidence intervals. Without DD, the TTS curves terminate at $n_{\max}^{'} = 16$ ( $n_{\max}^{'} = 20$ ) for Montreal (Cairo), since we find $p_{s} = 0$ for $n > n_{\max}^{'}$ . Moreover, $λ > 1$ without DD, indicating a worse-than-classical scaling. With DD protection, on Cairo, the $p_{s} > 0$ range is extended to $n = 23$ , and $λ$ is just below the breakeven point of 1, but the uncertainty is too large to conclude that quantum speedup has occurred. In contrast, the Montreal scaling with DD does exhibit quantum speedup. Since two-qubit operations and readout durations are shorter for Cairo, it exhibits a consistently lower absolute TTS than Montreal. We report $5 σ$ confidence intervals from bootstrapping for each data point; error bars are mostly covered by the symbols.
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Figure 4
Results for $λ_{h_{\max}}$ , the maximum local slope of each of the curves in Fig. 3 for $n \leq h_{\max}$ , i.e., the worst-case scaling when Fig. 3 is restricted to $h_{\max} + 1$ qubits. Only Montreal with DD exhibits an unambiguous quantum speedup, with $λ_{h_{\max}}$ well below 1 for all $n \leq h_{\max}$ . Error bars represent $2 σ$ confidence intervals.
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