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Experimental Superposition of Orders of Quantum Gates
Lorenzo M. Procopio, Amir Moqanaki, Mateus Araújo, Fabio Costa, Irati A. Calafell, Emma G. Dowd, Deny R. Hamel, Lee A. Rozema, Časlav Brukner, Philip Walther
TL;DR
The paper asks whether commuting and anticommuting unitary gates can be distinguished with one use of each under experimentally realizable conditions. It compares a fixed-order formulation with an experimentally implemented protocol and reports higher success for the experiment than for fixed-order benchmarks.
Problem
The task is to distinguish whether two otherwise unknown unitary gates commute or anticommute using one copy of each gate in a fixed-order circuit.
Method
The analysis represents fixed-order circuits with operators for the unitaries, measurement outcomes, and their connecting circuit, then optimizes success using a semidefinite program.
Results
0.976 ± 0.015 was the measured experimental success probability, compared with 0.9390 for the optimal fixed-order circuit on the 100 experimental gate pairs.
Takeaways & Limitations
The experimental protocol’s measured success probability exceeded the fixed-order benchmarks considered for this task.
Abstract
from arXiv · showhide
In a quantum computer, creating superpositions of quantum bits (qubits) in different states can lead to a speed-up over classical computers [1], but quantum mechanics also allows for the superposition of quantum circuits [2]. In fact, it has recently been theoretically predicted that superimposing quantum circuits, each with a different gate order, could provide quantum computers with an even further computational advantage [3-5]. Here, we experimentally demonstrate this enhancement by applying two quantum gates in a superposition of both possible orders to determine whether the two gates commute or anti-commute. We are able to make this determination with only a single use (or query) of each gate, while all quantum circuits with a fixed order of gates would require at least two uses of one of the gates [3]. Remarkably, when the problem is scaled to N gates, creating a superposition of quantum circuits is likely to provide an exponential advantage over classical algorithms, and a linear advantage over quantum algorithms with fixed gate order [4]. The new resource that we exploit in our experiment can be interpreted as a "superposition of causal orders". We demonstrate such a superposition could allow some quantum algorithms to be implemented with an efficiency that is unlikely to be achieved on a quantum computer with a fixed gate order.
A. Experimental Details
The experiment uses a photon-based interferometric setup to implement and characterize the superposition of gate orders. Measurements correct for unequal detection efficiencies and quantify interferometer stability and success.
- A Sagnac-loop source generated photon pairs using spontaneous parametric down-conversion in a 20 mm PPKTP crystal pumped by a 23.7 mW, 395 nm diode laser.
- Photon outputs were coupled into single-mode fibers and detected with avalanche photodiodes connected to an FPGA-based coincidence counter.
- Output probabilities were corrected using an efficiency factor η, typically around 0.7, to account for unequal coupling and detector efficiencies.
- The interferometer achieved visibility of 99.4 ± 0.2, with phase drift below 9 mrad per minute and negligible error over a two-minute measurement.
- Anti-commuting gates were generated as A1 = RσzR† and A2 = RσyR†, with waveplate angles computed for the target gates.
B. A Fixed-Order Quantum Circuit for Our Task
The appendix formulates the best fixed-order strategy as an optimization over circuit operators and compares its success probability with the experiment. The experimentally measured protocol outperformed fixed-order benchmarks.
- The fixed-order comparison evaluates circuits using one copy of each unitary and a single-qubit measurement, with success averaged over the two promised cases.
- Any candidate circuit is decomposed into an operator representing the unitaries and measurement outcome, together with W representing the connecting fixed-order circuit.
- The probability of measurement outcome i after applying U1 and U2 is represented by p(i|U1, U2).
- The success probability depends on the measures over commuting and anticommuting unitary pairs; Pauli-only sampling would give psucc = 1.
- 0.976 ± 0.015 was the experimentally measured success probability, exceeding the maximal success probability of any fixed-order circuit.
- 0.9390 was the optimal fixed-order circuit’s success rate averaged over the 100 gate pairs used experimentally.
C. Additional Raw Data
Additional tests examined input-state dependence and evaluated the protocol on 100 further gate pairs. The protocol maintained high average success across these tests.
- Changing qubit 2’s input state by rotating the preparation half-waveplate produced no significant performance changes across tested states.
- 0.976 ± 0.015 was the average success rate across 50 commuting and 50 anticommuting gate pairs.
D. Waveplate Settings
Two unitary gates were implemented with six motorized waveplates, while randomized gate pairs and Pauli gates were tested experimentally. The measurements achieved high average success rates for identifying whether gates commute or anti-commute.
- Waveplate implementation: Each unitary gate used three waveplates—quarter, half, then quarter—requiring six waveplates for two gates.Motorized rotation mounts allowed remote angle setting without disturbing interferometer phase.
- Pauli-gate settings: Table I specifies the quarter–half–quarter waveplate angles for implementing four Pauli gates as U1 and U2.The first three angle columns correspond to U1, and the final three correspond to U2.
- Pauli-gate settings: Different waveplate angles were used for σy and σz when implementing U1 and U2, without changing their logical gates.This also verifies that protocol success is independent of the physical apparatus used to enact each gate.
- Random-gate settings: The 100 tested random commuting and anti-commuting gate pairs were generated from a random gate R and implemented using tabulated waveplate settings.The anti-commuting gates were constructed as A1 = RσzR† and A2 = RσyR†.
- Experimental results: 0.970 ± 0.024 average success was obtained for Pauli-gate tests across several input states.The data identify commuting cases through photon exit at port 0 and anti-commuting cases through exit at port 1.
- Experimental results: 0.976 ± 0.015 average success was obtained for 100 pairs of randomly chosen unitary gates.The plotted cases distinguish anti-commuting pairs Ai from commuting pairs Ci.