Multivariate trace estimation in constant quantum depth

TL;DR

This work shows that estimating the multivariate trace $\operatorname{Tr}[\rho_1 \cdots \rho_m]$ can be achieved with a constant-depth quantum circuit using a two-dimensional nearest-neighbor architecture. The approach centers on a GHZ-based control scheme and a depth-two decomposition of the cyclic permutation, enabling highly parallelized operations and a practical estimator $\hat{T}=\hat{R}+i\hat{J}$ for $\operatorname{Tr}[\rho_1 \cdots \rho_m]$. The authors provide rigorous statistical guarantees, quantify sample and gate requirements, and extend the method to nonlinear functions of density matrices via polynomial approximations with explicit resource bounds. They also connect the framework to practical applications, including estimating functions like $\operatorname{Tr}[g(\rho)]$ and certain Schatten distances, highlighting potential near-term impact for quantum hardware such as Google's Sycamore-like architectures.

Abstract

There is a folkloric belief that a depth-$Θ(m)$ quantum circuit is needed to estimate the trace of the product of $m$ density matrices (i.e., a multivariate trace), a subroutine crucial to applications in condensed matter and quantum information science. We prove that this belief is overly conservative by constructing a constant quantum-depth circuit for the task, inspired by the method of Shor error correction. Furthermore, our circuit demands only local gates in a two dimensional circuit -- we show how to implement it in a highly parallelized way on an architecture similar to that of Google's Sycamore processor. With these features, our algorithm brings the central task of multivariate trace estimation closer to the capabilities of near-term quantum processors. We instantiate the latter application with a theorem on estimating nonlinear functions of quantum states with "well-behaved" polynomial approximations.

Figures (3)

• Figure 1: A constant-depth quantum circuit for preparing an eight-party GHZ state, assisted by measurement, classical feedback, and qubit resets. Method 1 consists of all the steps depicted, except for the qubit resets and final controlled-NOTs (but only prepares a five-party state). In Method 2, the measured qubits are additionally reset to the $|0\rangle$ state and connected by controlled-NOTs so that a larger, eight-party state can be prepared instead.
• Figure 2: The leftmost part of the circuit prepares a four-party GHZ state. The middle part of the circuit performs a controlled cyclic-shift. The final part of the circuit results in the classical bits $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$, which are used to generate $r=\left( -1\right) ^{x_{1}+x_{2}+x_{3}+x_{4}}$. As argued in Section \ref{['sec:guarantees']}, the expectation of $r$ is equal to $\operatorname{Re}[\operatorname{Tr}[\rho_{1}\cdots\rho_{8}]]$, so that this latter quantity can be estimated through repetition.
• Figure 3: (1) The squares in light grey represent control qubits, and the squares in dark grey represent data qubits. In this example, there are five data states involved, each consisting of four qubits. (2) The quantum data is loaded during this stage, which we note here can be conducted in parallel with the preparation of the GHZ state in the control qubits. The state $\rho_i$, for $i \in \{1, \ldots, 5\}$, is a four-qubit state that occupies the indicated column of the data qubits in dark grey. The light grey control qubits are prepared in the all zeros state. (3) First step of the preparation of the GHZ state of the control qubits. Every other control qubit has a Hadamard gate applied in parallel. (4) Every pair of control qubits has CNOT gates applied in parallel. (5) Every other pair of control qubits has CNOT gates applied in parallel. (6) Starting from the third control qubit from the top left, every other control qubit is measured, and the measurement outcome is stored in a binary vector $b_1, \ldots, b_7$. (7) Based on the measurement outcomes from the previous step, Pauli-$X$ corrections are applied to every other qubit, starting from the fourth in the top row. The particular corrections needed are abbreviated by a multivariate function $f$, the details of which are available in \ref{['eq:ghz-prep-correction']}. (8) The measured qubits are reset to the all zeros state. (9) Final step of the preparation of the GHZ state of the control qubits. CNOT gates are again applied to every other control qubit. The final state of all control qubits is equal to a GHZ state. (10) Controlled-SWAPs are applied in parallel between control qubits and data qubits, in a first round of the implementation of the cyclic shift. (11) Controlled-SWAPs are applied in parallel between other control qubits and data qubits, in a second round of the implementation of the cyclic shift. (12) Hadamard gates are applied to all control qubits. (13) In a final step, all control qubits are measured in the computational basis and the measurement outcomes are processed according to \ref{['eq:real-part-estimator']} to form an estimate of the real part of $\operatorname{Tr}[\rho_1 \cdots \rho_5]$. To estimate the imaginary part, replace every $H$ in step (12) with $HS^{\dag}$.

Theorems & Definitions (5)

• Lemma 1: Hoeffding H63
• Proposition 2
• Theorem 3
• Proposition 4
• Theorem 5