Lower bounds on the complexity of preparing mixed states

TL;DR

This work introduces a purification-based, lightcone-aware framework for lower-bounding the circuit depth required to prepare mixed quantum states. By linking correlations within a mixed state to the depth of an ensemble of geometrically local circuits, it derives exact and approximate bounds (Theorems Exact and Approx) that hold even when local ancilla qubits are allowed. The method is applied to Gibbs states of one-dimensional critical systems described by conformal field theory, revealing that the required depth diverges as temperature decreases at criticality up to a preparation-error cutoff. The results provide a general, correlation-driven tool for assessing mixed-state preparation complexity and offer avenues for extensions to measurement-feedback, ancilla-assisted schemes, and mixed-state topological order.

Abstract

We establish a relationship between the correlations in a many-qubit mixed state and the minimum circuit depth needed for its preparation. If the mutual information between two subsystems exceeds the mutual information between one of those subsystems and the environment, which purifies the mixed state of the system, then the past lightcones of the subsystems must intersect one another. This results in a lower bound on the circuit depth of any ensemble of geometrically local unitaries that prepares the state to some specified degree of approximation. As an application, we derive lower bounds on the circuit depth needed to prepare thermal states of one-dimensional quantum critical systems described by conformal field theory, showing that the depth diverges as temperature is decreased up to a cutoff set by the preparation error.

Figures (2)

• Figure 1: Starting from the initial state $\ket{0^{\otimes n}}$, a mixed state $\rho$ can be generated by applying randomly choosing unitaries $U_z$ with probabilities $p_z$. From such a preparation scheme, a purification can be constructed [Eq. \ref{['eq:PurificationDef']}], which can be viewed as the result of applying a $z$-controlled unitary $V = \sum_z \ket{z}\bra{z}_E \otimes U_z$, with the control $E$ initialized in the state $\sum_z \sqrt{p_z}\ket{z}_E$, and the system as the target. Dividing the system qubits into regions $A$,$B$,$C$, we denote $x_{AB}$ as the minimum graph distance between $A$ and $B$ (here a one-dimensional chain is shown). If each $U_z$ is a circuit of depth $d < \lfloor x_{AB}/2\rfloor + 1$, then the past lightcones of $A$ and $B$ (red and blue shaded gates, respectively) do not overlap, and $I(A:B)_{\phi_z} = 0$. The classical-quantum state $\rho^{ABCR}_{\rm opt}$ [Eq. \ref{['eq:RhoCQ']}] can be generated by applying a dephasing channel $\mathcal{E}^{E \rightarrow R}$ to $E$.
• Figure 2: Lower bounds on the circuit depth needed to prepare thermal states $\rho \propto e^{-\beta H}$ of the transverse field Ising chain, $H = -\sum_j( Z_j Z_{j+1} + g X_j)$. Here we use open boundary conditions, the length $n$ of the chain is $n=301$, $A$ is the central site, and $B$ consists of the first $(n+1)/2-x_{AB}$ sites of the chain. The channel $\mathcal{N}^{A \to A'}$ is a projective measurement of the Pauli-$X$ operator, so Eq. \ref{['eq:KeyBound']} applies. (a) Ratio $\chi_B/\chi_E$ at the quantum critical point, $g=1$, for various $x_{AB}$ and $\beta=10,20,\ldots,100$ (light to dark). Inset: Decay of $\chi_E$ (solid) at $g=1$ on increasing $\beta$, compared with the expected behavior $\sim \beta^{-2}$ (dotted). (b) Lower bounds on circuit depth for (light) exact and (dark) $\epsilon$-approximate state preparation for various transverse field strength $g$ (different colors, see legend). The bounds for approximate state preparation come from the condition $K \leq k(\epsilon)$ with $k(\epsilon) = 10^{-5}$, corresponding to $\epsilon \approx 10^{-7}$.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Definition 1: Local ancilla-assisted circuits
• Lemma 1
• Lemma 2