[Paper Review] Learning marginals suffices!

Nengkun Yu, Tzu-Chieh Wei|arXiv (Cornell University)|Mar 15, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper demonstrates that quantum states with low circuit complexity are uniquely determined by their low-order marginals—specifically, $2^D$-local reduced density matrices for depth-$D$ circuits—enabling efficient tomography with only Pauli measurements. The key result is that such states can be reconstructed using exponentially fewer samples than traditional tomography, breaking the exponential sample complexity barrier even in noisy settings.

ABSTRACT

Beyond computer science, quantum complexity theory can potentially revolutionize multiple branches of physics, ranging from quantum many-body systems to quantum field theory. In this paper, we investigate the relationship between the sample complexity of learning a quantum state and the circuit complexity of the state. The circuit complexity of a quantum state refers to the minimum depth of the quantum circuit necessary to implement it. We show that learning its marginals for the quantum state with low circuit complexity suffices for state tomography, thus breaking the exponential barrier of the sample complexity for quantum state tomography. Our proof is elementary and overcomes difficulties characterizing short-range entanglement by bridging quantum circuit complexity and ground states of gapped local Hamiltonians. Our result, for example, settles the quantum circuit complexity of the multi-qubit GHZ state exactly.

Motivation & Objective

• To resolve the long-standing challenge of high sample complexity in quantum state tomography for low-complexity quantum states.
• To establish a theoretical foundation linking quantum circuit complexity and the unique determinism of quantum states by their marginals.
• To develop a robust, noise-tolerant method for reconstructing low-complexity quantum states using only local Pauli measurements.
• To provide a practical and efficient alternative to full state tomography for near-term quantum devices (NISQ era).
• To offer a new data structure—tuples of reduced density matrices—for representing quantum states efficiently in classical simulations and certification.

Proposed method

• Introduce and extend quantum overlapping tomography to learn a set of reduced density matrices using random Pauli measurements.
• Prove that the $2^D$-local marginals uniquely determine the output state of a depth-$D$ quantum circuit, leveraging connections between circuit complexity and gapped local Hamiltonians.
• Use tools from quantum information theory, including perfect hash families and classical shadows, to bound the number of required Pauli measurements.
• Establish robustness of the reconstruction by showing that any state with similar marginals to a low-complexity state must be close in trace distance.
• Apply techniques from quantum many-body physics, such as short-range entanglement and ground state properties of gapped Hamiltonians, to characterize the structure of low-complexity states.
• Derive improved bounds on the number of marginals needed for unique reconstruction on structured lattices (e.g., 1D chains and square lattices), using combinatorial functions like $\gamma_2(D) \leq D^2 + (D+1)^2$.

Experimental results

Research questions

• RQ1 Can quantum states with low circuit complexity be uniquely reconstructed from their low-order marginals?
• RQ2 What is the minimal number of local measurements (e.g., Pauli) required to reconstruct such states, and does this scale exponentially or sub-exponentially?
• RQ3 How robust is the reconstruction procedure under noise or perturbations in the measured marginals?
• RQ4 Can the framework be extended to structured quantum systems such as 1D chains or 2D lattices with improved sample complexity bounds?
• RQ5 What are the implications of this result for quantum circuit optimization, certification, and the design of efficient data structures in quantum computing?

Key findings

• The output state of a depth-$D$ quantum circuit is uniquely determined by its $2^D$-local reduced density matrices, even on general interaction graphs.
• On a 1D chain, the state is uniquely determined by its $2D$-local marginals, significantly improving the bound compared to general graphs.
• On square lattices, the required locality is bounded by $\gamma_2(D) \leq D^2 + (D+1)^2$, providing a structured improvement over the general case.
• The reconstruction procedure is robust: any state with marginals close to those of a low-complexity state must be close in trace distance, enabling noise-tolerant certification.
• The method breaks the exponential sample complexity barrier for tomography of low-complexity states, even when only Pauli measurements are allowed.
• The result settles the exact quantum circuit complexity of the multi-qubit GHZ state, confirming it is $\Theta(n)$, and provides a theoretical justification for using reduced density matrices as a data structure in NISQ-era quantum computing.