Interactive shallow Clifford circuits: quantum advantage against NC$^1$ and beyond

Recent work of Bravyi et al. and follow-up work by Bene Watts et al. demonstrates a quantum advantage for shallow circuits: constant-depth quantum circuits can perform a task which constant-depth classical (i.e., AC$^0$) circuits cannot. Their results have the advantage that the quantum circuit is fairly practical, and their proofs are free of hardness assumptions (e.g., factoring is classically hard, etc.). Unfortunately, constant-depth classical circuits are too weak to yield a convincing real-world demonstration of quantum advantage. We attempt to hold on to the advantages of the above results, while increasing the power of the classical model. Our main result is a two-round interactive task which is solved by a constant-depth quantum circuit (using only Clifford gates, between neighboring qubits of a 2D grid, with Pauli measurements), but such that any classical solution would necessarily solve $\oplus$L-hard problems. This implies a more powerful class of constant-depth classical circuits (e.g., AC$^0[p]$ for any prime $p$) unconditionally cannot perform the task. Furthermore, under standard complexity-theoretic conjectures, log-depth circuits and log-space Turing machines cannot perform the task either. Using the same techniques, we prove hardness results for weaker complexity classes under more restrictive circuit topologies. Specifically, we give QNC$^0$ interactive tasks on $2 \times n$ and $1 \times n$ grids which require classical simulations of power NC$^1$ and AC$^{0}[6]$, respectively. Moreover, these hardness results are robust to a small constant fraction of error in the classical simulation. We use ideas and techniques from the theory of branching programs, quantum contextuality, measurement-based quantum computation, and Kilian randomization.

💡 Research Summary

The paper “Interactive shallow Clifford circuits: quantum advantage against NC¹ and beyond” builds on earlier results showing that constant‑depth quantum circuits can solve tasks that constant‑depth classical circuits (AC⁰) cannot. Those earlier separations, while unconditional, involve a classical model that is too weak to be convincing for real‑world quantum‑advantage demonstrations. The authors therefore introduce a more powerful classical model and an interactive protocol that preserves the practicality of the quantum side while dramatically raising the bar for any classical simulation.

Model and task.
The authors define a two‑round interactive task Tₘ,ₙ on an m×n qubit grid. In the first round the quantum device (or a classical simulator) receives measurement bases for all but a constant fraction of the qubits, executes a constant‑depth Clifford circuit on a 2‑dimensional lattice, and returns the measurement outcomes. In the second round the device is given bases for the remaining qubits and must return the corresponding outcomes. The quantum implementation uses only nearest‑neighbor Clifford gates and Pauli measurements, which can be realized on near‑term hardware. The crucial difference is that a classical simulator can “rewind” after the first round: it can keep a copy of the internal state and reuse it for many different second‑round measurements, a capability that a genuine quantum device does not possess because of the no‑cloning theorem.

Hardness strategy.
The authors’ hardness proofs follow a common template:

1. Encoding a hard decision problem into a Clifford sequence.
   A fixed sequence of Clifford gates is controlled by a classical input, exactly like a branching program. The output of the quantum circuit (the distribution of measurement outcomes) encodes the solution to a known hard problem. For two‑qubit Clifford gates this yields NC¹‑completeness via Barrington’s theorem (the group generated by two‑qubit Cliffords contains a copy of S₅). For a specially chosen CNOT‑only sequence on n qubits the problem becomes ⊕L‑hard (solving linear equations mod 2).

2. Flattening to constant depth via measurement‑based quantum computation (MBQC).
   The authors use the Raussendorf‑Briegel construction that any Clifford circuit can be performed by a depth‑1 MBQC pattern with only Pauli corrections. The corrections appear as Pauli operators that can be pushed to the end of the computation. In the interactive setting the first round creates the entangled resource state; the second round measures it in bases that reveal the hidden Pauli string.

3. Forcing the classical simulator to reveal the Pauli string.
   By embedding the Magic Square game (for NC¹) or the Magic Pentagram game (for ⊕L) into the measurement choices, the protocol guarantees that any classical simulator that returns a valid outcome must implicitly commit to a consistent Pauli string. Inconsistent strings would violate the perfect‑quantum‑strategy correlations of these contextuality games, causing the verifier to reject.

4. Random self‑reduction (Kilian randomization).
   The authors allow the classical simulator to err with bounded probability. They then apply a random self‑reduction: the verifier repeats the second‑round measurement with independently chosen bases, amplifying a modest success probability (≥½) to overwhelming confidence. This step shows that even error‑tolerant AC⁰ circuits (BP·AC⁰) cannot solve the task unless they can solve the underlying hard problem.

Main results.
The paper presents three parameterized hardness statements:

• m = 1 (1 × n grid). The task is AC⁰