⚛️ quantum physics

Learning State Preparation Circuits for Quantum Phases of Matter

This paper presents a flexible, classically efficient framework that uses local reduced density matrices to construct asymptotically optimal state preparation circuits for both short-range and long-range entangled many-body ground states by leveraging a robust variant of the quantum Markov chain condition.

Original author: Hyun-Soo Kim, Isaac H. Kim, Daniel Ranard

Published 2026-11-09

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CC BY 4.0

Original author: Hyun-Soo Kim, Isaac H. Kim, Daniel Ranard

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum computers promise to solve problems that are impossible for today's machines, particularly in simulating the behavior of matter itself. To do this, scientists must first teach the computer how to create a specific starting point: a complex arrangement of particles known as a quantum state. Imagine trying to study the weather, but you cannot begin your simulation until you have perfectly assembled the exact pattern of clouds and wind currents you wish to observe. In the quantum world, this starting pattern is often the lowest energy state of a system, or its "ground state." For decades, the standard way to prepare these states has relied on a method called quantum phase estimation, which works well only if the computer already starts with a guess that is very close to the target. If the initial guess is poor, the process fails. This creates a bottleneck: to simulate a new material, researchers often need a new, highly efficient way to build the starting state from scratch, a task that has proven difficult for many types of matter.

A team of researchers has now developed a flexible and efficient method to learn how to build these complex starting states, even when they possess intricate, long-range connections that have previously defied simple description. Their approach does not require knowing the entire, global structure of the state beforehand. Instead, it works like a puzzle solver that only looks at small, local pieces. By examining the reduced density matrices—essentially the statistical snapshots of small, ball-shaped regions of the system—their algorithm can deduce the exact sequence of operations needed to assemble the entire system. This method applies to two broad categories of quantum matter. The first includes states that can be built quickly, such as those formed by short-range interactions. The second, more challenging category includes states with long-range entanglement, like the toric code, where particles are linked across vast distances in ways that cannot be created by simple, shallow circuits.

The researchers proved that for these specific types of matter, their algorithm can find a preparation circuit that is not only correct but also asymptotically optimal, meaning it uses the minimum possible depth of operations required by the laws of physics for that system. For states with long-range entanglement in two dimensions, the algorithm finds a circuit whose depth scales with the square root of the number of particles, which is the best possible outcome. The core of their discovery relies on a concept called local extendibility. This property allows the researchers to take the information from a small region and mathematically "extend" it to a slightly larger region, effectively stitching together local pieces to form the whole. Crucially, they showed that this property remains stable even if the state has been altered by a constant-depth circuit, making the method robust against the kinds of transformations that often break other approaches.

The process begins by gathering data from the quantum state, either from a theoretical description or from a physical quantum simulator. The algorithm then performs a classical calculation to learn the local "glue" that connects these small regions. It checks if the state satisfies specific conditions that allow for this local extension. If the conditions are met, the algorithm constructs a sequence of operations that builds the global state layer by layer. If the state does not fit the criteria, the algorithm simply fails to find a circuit, but it never produces a wrong answer; it either finds the correct preparation method with a rigorous guarantee or it stops. This reliability means the method can be used as a black box to convert various descriptions of quantum matter into executable circuits.

This work opens a direct path to using near-term quantum simulators to learn how to prepare complex states. Since local measurements can be taken from these simulators, the algorithm can learn the preparation circuit directly from the experiment, bypassing the need for a perfect theoretical model. It also offers a way to compress existing descriptions of quantum states into more efficient circuits. While the method currently applies to specific classes of matter, such as invertible states in any dimension and certain topologically ordered states in two dimensions, it represents a significant step toward automating the setup of quantum simulations. By turning the difficult problem of state preparation into a tractable learning task based on local information, the researchers have provided a powerful new tool for exploring the quantum phases of matter.

Technical Summary: Learning State Preparation Circuits for Quantum Phases of Matter

Problem Statement

The simulation of physical systems on quantum computers requires efficient preparation of many-body ground states. While dynamics simulation has seen significant algorithmic advances (e.g., qubitization, linear combination of unitaries), preparing the initial low-energy state remains a bottleneck. Traditional methods like quantum phase estimation require an initial state with high overlap with the target ground state, necessitating a method to find a low-complexity circuit that prepares the state.

Existing approaches often rely on efficient classical descriptions of the state (e.g., tensor networks, stabilizer states) or assume the state is preparable by a constant-depth circuit. A recent method [18] demonstrated that state preparation circuits could be learned from reduced density matrices (RDMs) over $O(1)$ -sized subsystems, but it was restricted to one and two dimensions and only applied to states preparable by constant-depth circuits without ancillas. This excludes important physical phases, such as topologically ordered states (e.g., the toric code), which cannot be prepared by constant-depth circuits and require circuit depths scaling with system size (e.g., $\Omega(\sqrt{n})$ in 2D).

This paper addresses the challenge of learning state preparation circuits for a broader class of quantum phases, including:

Short-range entangled (invertible) states in arbitrary dimensions, potentially involving ancillary qubits.
Long-range entangled states (specifically those with "liquid" topological order) in two dimensions, such as the toric code and string-net models, provided they are unique ground states (e.g., on a sphere) or subsystems thereof.

Methodology

The core theoretical framework relies on the concept of locally extendible states, a relaxation of the quantum Markov chain condition.

1. Local Extendibility and Recovery

A state is defined as locally extendible if its reduced density matrix over a specific partition of subsystems (e.g., $BCDE$ ) can be recovered from a smaller partition (e.g., $BE$ ) via a quantum channel. Specifically, for a purification $\rho_{ABCDE}$ , a channel $\Phi: \mathcal{D}_{BE} \to \mathcal{D}_{BC}$ exists such that $\rho_{ABC} = I_A \otimes \Phi(\rho_{ABE})$ .

Robustness: Unlike the standard quantum Markov chain condition, which is fragile under constant-depth circuits, the property of local extendibility (specifically for the partitions defined in the paper) is proven to be stable under geometrically local constant-depth circuits, provided the subsystems are sufficiently large relative to the circuit depth.
Verification: The authors provide a polynomial-time classical algorithm to verify extendibility and find the extending channel. This is formulated as a semi-definite programming (SDP) problem maximizing the fidelity of recovery. Alternatively, if the circuit architecture of the extending map is known, a brute-force search over gates can be performed.

2. Circuit Construction via Stitching

The global state preparation circuit is constructed by "stitching" together local extending maps.

Short-range entangled states: The global state is built by sequentially applying extending maps that connect disjoint regions, effectively growing the state from local patches to the global system. This results in a constant-depth circuit (or $O(kd)$ in $k$ dimensions).
Long-range entangled states: For states like the toric code, the construction follows a sequential pattern (similar to a sequential circuit) that builds the state layer by layer. This results in a circuit depth scaling as $O(\sqrt{n})$ in 2D, which is asymptotically optimal for geometrically local circuits.

3. Learning Algorithm

The learning algorithm operates as follows:

Tomography: Perform local state tomography on $O(1)$ -sized balls to obtain approximate reduced density matrices.
Extension Learning: Use the SDP (or brute-force) method to learn the extending channels for these local RDMs.
Error Accumulation: The algorithm iteratively increases the radius of the local balls if the accumulated error exceeds the target.
Composition: The learned extending maps are composed to form the global state preparation circuit.
Certification: If the local conditions (extendibility) are satisfied, the resulting circuit is guaranteed to prepare the target state within a specified trace distance. If the state does not belong to the target class, the algorithm may fail to find a circuit in polynomial time, but any circuit found is rigorously certified.

Key Contributions and Results

1. Generalization to Arbitrary Dimensions and Ancillas

The authors present an algorithm for invertible states (states preparable by constant-depth circuits with ancillas) in any number of spatial dimensions $k$ .

Theorem 1: Given samples of an $n$ $n$ -qubit invertible state prepared by a depth- $d$ $d$ circuit, the algorithm learns a circuit with trace distance $\epsilon$ $ϵ$ .
- Complexity: It can output a circuit of depth $O(ke^{O(dk)})$ in time $\tilde{O}(e^{O(dk)}n^3/\epsilon^2)$ , or a circuit of depth $O(kd)$ in time $O((n/\epsilon)^{O(dk)})$ .
- Significance: This generalizes prior work [18] to higher dimensions and the more general class of invertible states (including those requiring ancillas, such as Kitaev's $E_8$ state).

2. Learning Long-Range Entangled States in 2D

The authors present an algorithm for a subclass of long-range entangled states (liquid topological order) in two dimensions, including the toric code and string-net models on a disk with uniform boundary conditions.

Theorem 2 (Informal): Given samples of an $n$ $n$ -qubit state formed by applying a depth- $d$ $d$ circuit to a long-range entangled state satisfying specific local conditions, the algorithm learns a geometrically local circuit.
- Scope: The result applies specifically to states that are unique ground states (e.g., the toric code on a sphere) or subsystems of such states. The algorithm's specific input requirement is a state on a disk with a uniform boundary condition (e.g., the toric code ground state on a disk with a smooth boundary condition). It does not apply to ground state subspaces with degeneracy where locally indistinguishable states exist (e.g., toric code with mixed smooth and rough boundaries).
- Complexity: It can output a circuit of depth $O(\sqrt{n}e^{O(d^2)})$ in time $\tilde{O}(n^3/\epsilon^2)$ , or a circuit of depth $O(d\sqrt{n})$ in time $O((n/\epsilon)^{O(d^2)})$ .
- Optimality: The $\sqrt{n}$ depth scaling is proven to be asymptotically optimal for geometrically local circuits preparing topologically ordered ground states.

3. Robustness and Certification

The method relies on the stability of local extendibility under constant-depth circuits. This allows the algorithm to learn circuits for states that are perturbations of ideal phases. Furthermore, the algorithm provides a rigorous guarantee: if it outputs a circuit, that circuit prepares the state with the promised accuracy, regardless of whether the input state was known to belong to the target class a priori.

Significance and Claims

The paper claims to provide a flexible and efficient framework for obtaining state preparation circuits for a large class of many-body ground states directly from local reduced density matrices.

Bridging Theory and Experiment: The method enables learning state preparation circuits directly from quantum simulators or quantum computers using classical shadows, without needing a full classical description of the global wavefunction.
Handling Topological Order: It successfully addresses the learning problem for topologically ordered states (long-range entangled), which were previously out of reach for constant-depth circuit learning methods, subject to the constraint of unique ground states or their subsystems.
Classical Tractability: The optimization landscape for finding the extending maps is classically tractable (via SDP), avoiding the exponential complexity associated with learning arbitrary quantum states.
Optimality: For long-range entangled states, the learned circuits achieve the asymptotically optimal depth scaling ( $\Omega(\sqrt{n})$ ) required by locality constraints.

The authors note that their approach is limited to "liquid" topological phases and does not currently cover fracton phases or states with degenerate ground subspaces that are locally indistinguishable. They also leave the extension to quasi-local unitaries and adaptive circuits as open problems for future work.