Quantum Advantage from Sampling Shallow Circuits: Beyond Hardness of Marginals

TL;DR

The paper establishes an unconditional quantum advantage for sampling by constructing a uniform family of depth-7 quantum circuits that exactly sample a target distribution while strong classical NC^0 samplers remain far from this distribution in total variation distance. Central to the result is a carefully designed host/hard distribution pair, derived from the Parity Halving Problem and a tree-based relational construction, linked by a 5-local reduction. The authors prove a robust NC^0 lower bound via a structured analysis and then amplify the separation using a direct-product theorem to achieve a distance of $1-e^{-\Omega(n)}$ for large instances. The work also analyzes the QNC^0 upper bound, showing how a shallow, geometrically local circuit can realize the host distribution, and discusses gate-set minimality and potential extensions to AC^0 versus QNC^0 separations, highlighting both theoretical significance and practical pathways for quantum advantage experiments.

Abstract

We construct a family of distributions $\{\mathcal{D}_n\}_n$ with $\mathcal{D}_n$ over $\{0, 1\}^n$ and a family of depth-$7$ quantum circuits $\{C_n\}_n$ such that $\mathcal{D}_n$ is produced exactly by $C_n$ with the all zeros state as input, yet any constant-depth classical circuit with bounded fan-in gates evaluated on any binary product distribution has total variation distance $1 - e^{-Ω(n)}$ from $\mathcal{D}_n$. Moreover, the quantum circuits we construct are geometrically local and use a relatively standard gate set: Hadamard, controlled-phase, CNOT, and Toffoli gates. All previous separations of this type suffer from some undesirable constraint on the classical circuit model or the quantum circuits witnessing the separation. Our family of distributions is inspired by the Parity Halving Problem of Watts, Kothari, Schaeffer, and Tal (STOC, 2019), which built on the work of Bravyi, Gosset, and König (Science, 2018) to separate shallow quantum and classical circuits for relational problems.

Figures (2)

• Figure 1: On the left is a $\mathsf{QNC^0}/\mathsf{qpoly}$ circuit which solves the Parity Halving Problem and on the right is a $\mathsf{QNC^0}$ circuit which solves the Relaxed Parity Having Problem over a graph $G = (V, E)$. Here $U_G$ is the ($|V| + |E|$)-qubit unitary which acts as $U_G\ket{z}\ket{b} = \ket{z}\bigotimes_{e = (u, v) \in E}\ket{b_e\oplus z_u \oplus z_v}$ for all $z \in \{0, 1\}^V$ and $b \in \{0, 1\}^E$.
• Figure 2: A depiction of the $\mathsf{QNC^0}$ circuit whose measurement distribution on the last $3n - 1$ qubits is exactly $\mathcal{D}_{\mathsf{host}}(\mathcal{T})$. Here $U_{\mathcal{T}}$ is the $(2n - 1)$-qubit unitary which acts as $U_{\mathcal{T}}\ket{z}\ket{b} = \ket{z}\bigotimes_{e = (u, v)\in E}\ket{b_e\oplus z_u \oplus z_v}$ - as shown in the proof of \ref{['prop:qnc0']}$U_{\mathcal{T}}$ can be implemented via $\mathsf{CNOT}$s in depth $2\Delta$. While the circuit shown above is not geometrically local, a rearrangement of the qubit wires would allow for all $\mathsf{CNOT}$, $\mathsf{Tof}$, and $\mathsf{CS}$ gates to act only on adjacent qubits in a $2D$-grid architecture.

Theorems & Definitions (48)

• Theorem 1.1: Informal Version of \ref{['thm:main_quantitative']}
• Theorem 2.1
• Remark 2.2
• Definition 2.3: Parity Halving Problem
• Definition 2.4: Relaxed Parity Halving Problem
• Definition 2.5: The $\mathcal{D}_{\mathsf{host}}(\mathcal{T})$ Distribution
• Proposition 2.5
• Definition 2.6: The $\mathcal{D}_{\mathsf{hard}}(n,m)$ Distribution
• Lemma 2.6
• Theorem 2.7