[Paper Review] Experimental demonstration of the advantage of adaptive quantum circuits

Michael Foss‐Feig, Arkin Tikku+17 more|arXiv (Cornell University)|Feb 6, 2023

Quantum Computing Algorithms and Architecture17 citations

TL;DR

The paper experimentally shows that a constant-depth adaptive quantum circuit with mid-circuit measurements and feed-forward can prepare a toric code ground state with fidelity 76.9% in depth 4, outperforming any purely unitary circuit of the same depth which is upper-bounded at 50%.

ABSTRACT

Adaptive quantum circuits employ unitary gates assisted by mid-circuit measurement, classical computation on the measurement outcome, and the conditional application of future unitary gates based on the result of the classical computation. In this paper, we experimentally demonstrate that even a noisy adaptive quantum circuit of constant depth can achieve a task that is impossible for any purely unitary quantum circuit of identical depth: the preparation of long-range entangled topological states with high fidelity. We prepare a particular toric code ground state with fidelity of at least $76.9\pm 1.3\%$ using a constant depth ($d=4$) adaptive circuit, and rigorously show that no unitary circuit of the same depth and connectivity could prepare this state with fidelity greater than $50\%$.

Motivation & Objective

• Motivate and demonstrate that adaptive quantum circuits can reduce quantum resource requirements by leveraging classical processing.
• Show that constant-depth adaptive circuits can prepare long-range entangled states beyond unitary-only capabilities.
• Provide a verifiable fidelity metric distinguishing adaptive from non-adaptive circuits under the same hardware constraints.
• Implement and test the protocol on a trapped-ion quantum computer to validate theoretical bounds.

Proposed method

• Prepare an initial product state of data qubits in |+> and measure Z-type stabilizers using ancilla qubits to define a geometry.
• Apply classical feed-forward by conditioning X corrections on Z-stabilizer measurement outcomes to purge defects and reach the ground state manifold.
• Use a depth-4 adaptive circuit to prepare the logical |+> state of the rotated surface code with open boundaries.
• Estimate fidelity via two-basis measurements to bound the overlap with the target logical state.
• Compare to a depth-4 local-unitary circuit bound showing fidelity cannot exceed 50% for the same geometry and depth.

Experimental results

Research questions

• RQ1 Can a constant-depth adaptive circuit prepare a long-range entangled toric code ground state with fidelity exceeding any constant-depth local unitary circuit within the same geometry?
• RQ2 What fidelity bounds can distinguish adaptive feed-forward from purely unitary circuits under identical resource constraints?
• RQ3 How does mid-circuit measurement and real-time classical processing enable preparation of topologically ordered states within fixed depth?

Key findings

• Adaptive depth-4 circuit achieves fidelity F ≥ 0.769 with a ±0.013 uncertainty for the toric code ground state.
• An upper bound shows any depth-4 local-unitary circuit on the same geometry cannot achieve fidelity above 0.5.
• Experimental implementation used 19 qubits (12 data, 7 ancilla) on Quantinuum H1-1 with real-time classical logic.
• Fidelity estimation uses P_x and P_z projections from X- and Z-basis measurements to bound F via F ≥ ⟨P_x⟩ + ⟨P_z⟩ − 1.
• The result demonstrates a verifiable advantage of adaptive circuits over non-adaptive circuits with identical depth and connectivity.