Sample-optimal single-copy quantum state tomography via shallow depth measurements

Technical Summary: Sample-optimal single-copy quantum state tomography with shallow-depth measurements

Problem Statement
Quantum state tomography (QST) is the task of reconstructing a quantum state $\rho$ from measurement data. While multi-copy collective measurements can achieve optimal sample complexity, they are experimentally infeasible on near-term noisy quantum devices due to the lack of error correction and limited coherence. Consequently, single-copy measurement strategies are preferred. However, existing single-copy protocols that achieve near-optimal sample complexity typically require deep circuits (depth $O(n)$) or complex positive-operator valued measurements (POVMs), which are impractical for current hardware. The central challenge addressed in this work is whether shallow-depth local circuits (specifically depth $O(\log n)$) can achieve sample-optimal QST for single-copy measurements without sacrificing reconstruction accuracy.

Methodology
The authors propose an ancilla-free single-copy QST protocol utilizing local random quantum circuits of logarithmic depth. The methodology relies on the following components:

1. Unitary Ensembles: Two specific ensembles of shallow-depth circuits are constructed using Clifford gates:
   • Two-layer block ensemble (Fig. 2b): A staggered pattern of two layers of $k$-qubit Clifford blocks ($U = U_2 U_1$), where $k = O(\log n)$. This is used for rank-$r$ states.
   • Single-layer block ensemble (Fig. 2a): Independent $k$-qubit Clifford blocks sampled from the Clifford group. This is used for full-rank states.
2. Estimator Construction: The protocol employs an unbiased estimator $\hat{\rho} = M^{-1}(U^\dagger |b\rangle\langle b| U)$, where $M$ is the shadow channel associated with the unitary ensemble. Unlike previous works that utilized biased estimators (sufficient for fidelity estimation but insufficient for trace-norm guarantees), this work constructs an unbiased estimator to ensure rigorous trace-distance bounds.
3. Concentration Analysis: Proving the concentration of the empirical average of the unbiased estimator is technically challenging. The authors combine Weingarten calculus with the transfer matrix method.
   • For rank-$r$ states, they utilize the matrix Bernstein inequality.
   • For full-rank states, they apply McDiarmid's inequality.
   • The transfer matrix technique is used to bound the second moment of the estimator by analyzing the coupling of Pauli operators across the block structure of the circuits.

Key Contributions and Results

• Rank-$r$ States: For a quantum state of rank $r$ in dimension $d=2^n$, the protocol using the two-layer block ensemble with depth $O(\log n)$ achieves a trace-norm error $\epsilon$ with sample complexity:
  $$T = O\left(\frac{d r^2 \log d}{\epsilon^2}\right)$$
  This matches the known single-copy lower bound $\Omega(dr^2/\epsilon^2)$ up to a logarithmic factor. The proof demonstrates that an overlap of $O(\log n)$ qubits between consecutive blocks is sufficient to maintain this scaling.

• Full-Rank States: For full-rank mixed states ($r=d$), the protocol using the simpler single-layer block ensemble achieves the optimal sample complexity scaling:
  $$T = O\left(\frac{d^3}{\epsilon^2}\right)$$
  This removes the logarithmic overhead present in the rank-$r$ case and matches the theoretical lower bound for full-rank states.

• Classical Post-Processing: The paper provides explicit algorithms for classical post-processing.

  • For rank-$r$ states, the runtime is $\tilde{O}(d^2 r^2 / \epsilon^2)$ with memory $O(d^2)$.
  • For full-rank states, the runtime is $\tilde{O}(d^2)$ with memory $O(d^2)$, which is optimal up to polylogarithmic factors since outputting a density matrix inherently requires $\Omega(d^2)$ time.

• Numerical Validation: Numerical simulations on random 8-qubit states confirm the theoretical predictions. The simulations show that shallow-depth circuits ($k=2$ for rank-$r$, $k \in \{1, 2, 4, 8\}$ for full-rank) achieve reconstruction errors comparable to global Clifford ensembles ($k=n$) as the number of samples increases.

Significance and Claims
The authors claim that their results demonstrate that sample-optimal QST can be realized using experimentally accessible shallow-depth measurements. Specifically:

1. Feasibility: The requirement for deep circuits ($O(n)$) is relaxed to $O(\log n)$, making the protocol viable for near-term quantum devices with limited connectivity and coherence.
2. Optimality: The protocol achieves the optimal sample complexity for full-rank states and near-optimal complexity for low-rank states, bridging the gap between theoretical lower bounds and practical experimental constraints.
3. Estimator Necessity: The work highlights that while biased estimators suffice for specific quantities like fidelity, unbiased estimators are crucial for guaranteeing small trace-norm errors in QST, necessitating the specific concentration proofs provided.

The paper modestly notes that for rank-$r$ states, it remains an open question whether the logarithmic factor $\ln(d)$ in the upper bound can be eliminated. Additionally, while the analysis relies on unitary ensembles capable of forming approximate unitary designs, the statistical proof does not explicitly depend on the approximate-design property, leaving open the question of whether the circuit architecture itself is the essential factor for QST performance.