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Decomposition of Matrix Product States into Shallow Quantum Circuits
Manuel S. Rudolph, Jing Chen, Jacob Miller, Atithi Acharya, Alejandro Perdomo-Ortiz
TL;DR
TN states can be transferred to PQCs, but efficient, accurate conversion into shallow circuits remains an open need for joint TN–PQC approaches. This work benchmarks analytic and optimization-based MPS decomposition protocols, finding sequential circuit growth with optimization most successful, especially under limited computational budgets.
Problem
Efficient, accurate conversion of TN states into shallow quantum circuits remains scarce, despite its importance to joint TN–PQC approaches.
Source: §I · p. 1Method
The authors benchmark MPS decomposition protocols combining analytic decomposition with constrained optimization of two-qubit circuit unitaries, including sequential circuit growth.
Source: §I · p. 2Results
Iter[DiOall] is the most successful protocol, delivering lower approximation errors than other methods with a larger advantage under limited computational budgets.
Source: §I · p. 2, §IV · p. 9, §V · p. 9Takeaways & Limitations
The protocol is a promising tool for mapping MPS to shallow circuits for joint applications of tensor networks and PQCs.
Source: §I · p. 2A rigorous derivation of the computational resources required for practical MPS decomposition is outside the work's scope.
Source: D. Discussion of Computational… · p. 7
Abstract
from arXiv · showhide
The rapid pace of recent advancements in numerical computation, notably the rise of GPU and TPU hardware accelerators, have allowed tensor network (TN) algorithms to scale to even larger quantum simulation problems, and to be employed more broadly for solving machine learning tasks. The "quantum-inspired" nature of TNs permits them to be mapped to parametrized quantum circuits (PQCs), a fact which has inspired recent proposals for enhancing the performance of TN algorithms using near-term quantum devices, as well as enabling joint quantum-classical training frameworks which benefit from the distinct strengths of TN and PQC models. However, the success of any such methods depends on efficient and accurate methods for approximating TN states using realistic quantum circuits, something which remains an unresolved question. In this work, we compare a range of novel and previously-developed algorithmic protocols for decomposing matrix product states (MPS) of arbitrary bond dimensions into low-depth quantum circuits consisting of stacked linear layers of two-qubit unitaries. These protocols are formed from different combinations of a preexisting analytical decomposition scheme with constrained optimization of circuit unitaries, and all possess efficient classical runtimes. Our experimental results reveal one particular protocol, involving sequential growth and optimization of the quantum circuit, to outperform all other methods, with even greater benefits seen in the setting of limited computational resources. Given these promising results, we expect our proposed decomposition protocol to form a useful ingredient within any joint application of TNs and PQCs, in turn further unlocking the rich and complementary benefits of classical and quantum computation.
I. INTRODUCTION
Tensor networks and parametrized quantum circuits have complementary strengths, motivating methods to transfer TN states into shallow, hardware-compatible circuits. This work benchmarks protocols for decomposing arbitrary-bond-dimension MPS into layers of two-qubit unitaries and finds sequential circuit growth with optimization most successful.
I. INTRODUCTION
- TNs support classical analysis and large-scale simulation, while PQCs can prepare states with long-range entanglement that TNs may not encode.TNs also provide a route to exact state-amplitude and fidelity calculations.Source: §I · p. 1
A. Tensor Networks
- Tensor networks are graph-structured factorizations that represent quantum states and have been used for quantum simulation, machine learning, and combinatorial optimization.Their classical scalability has benefited from GPU and TPU advances.Source: §I · p. 1, A. Tensor Networks · p. 2
I. INTRODUCTION
- Joint TN–PQC schemes depend on transferring TN states accurately into shallow circuits, since hardware noise can reduce fidelity in deep circuits.The introduction identifies a relative scarcity of computational techniques for this conversion and says best practices remain lacking.Source: §I · p. 1
- Sequential circuit growth with optimization yields lower approximation errors than the other benchmarked protocols, with a larger advantage under limited computational budgets.The protocols combine an analytical decomposition with constrained optimization of circuit unitaries.Source: §I · p. 2
A. Tensor Networks
- TN simulation can be effective when its graph matches the circuit layout, but mismatched layouts can require exponentially growing bond dimensions for high-fidelity simulation.Distant circuit gates can require entanglement to pass through intermediate TN edges.Source: A. Tensor Networks · p. 2
B. Canonical forms and Isometries
Canonical forms convert MPS tensors into isometries that can be extended to multi-qubit unitaries, yielding an exact staircase circuit representation. Because general multi-qubit gates are not directly supported by typical hardware, the work targets approximations using layers of two-qubit gates.
B. Canonical forms and Isometries
- Extending an isometry with orthonormal kernel-space columns produces a square unitary matrix.The added columns are orthogonal to the isometry's columns and to one another.Source: B. Canonical forms and Isometries · p. 2
- Canonical MPS tensors become isometries that can be converted into multi-qubit unitaries, arranged as one staircase layer of gates.Padding bond dimensions to powers of two and applying iterated QR decompositions yields the isometries.Source: B. Canonical forms and Isometries · p. 2
- Typical quantum hardware supports only one- and two-qubit gates, so arbitrary multi-qubit unitaries cannot be directly implemented.The work is motivated by efficient and accurate conversion from TNs to PQCs despite this restriction.Source: B. Canonical forms and Isometries · pp. 2–3
- For χmax = 2, an MPS maps exactly to one linear layer of two-qubit unitaries; general bond dimensions instead yield multi-qubit gates targeted for decomposition into two-qubit layers.The general bond-χ gate acts on ⌈log2(χ)⌉ + 1 qubits.Source: Figure 1 · p. 3
C. Related Work
Prior MPS-to-circuit approaches use analytic decomposition, unitary optimization, or both. This work builds on their complementary strengths and reports better ways to combine the two approaches.
C. Related Work
- Joint TN–PQC optimization was proposed for QML, while numerical work verified joint-training benefits for MPS with χmax = 2, which decompose into one two-qubit layer.Ref. [10] uses the conversion method described in this paper to initialize PQCs from larger TN models.Source: C. Related Work · p. 3
- Ref. uses iterated SVD truncations to initialize layers, saving resources but offering limited gains as depth increases and sometimes reducing fidelity beyond a critical depth.Ref. instead iteratively optimizes two-qubit gates, but can be vulnerable to barren plateaus without a high-quality initialization and costly when many sweeps are needed.Source: C. Related Work · p. 3
- Ref. [30] combines analytical gate initialization with iterative unitary optimization, and the authors confirm its advantage over Ref..They identify this as one benchmarked protocol, DallOall, while reporting that still better combinations exist.Source: C. Related Work · pp. 3–4
- The cited passage describes better ways of utilizing two styles of decomposition to approximate arbitrary matrix product states (MPS) using quantum circuits.Source: C. Related Work · p. 4
III. METHODS
This section introduces the algorithmic building blocks and decomposition protocols, with notation and indexing conventions specified in Appendix A.
III. METHODS
- The section outlines the algorithms and protocols for MPS decomposition and directs readers to Appendix A for unitary and layer notation.The notation covers U and L[U].Source: §III · p. 4
A. Analytic Decomposition by Disentangling
The analytic method approximates an MPS by repeatedly truncating it to bond dimension 2 and using the resulting two-qubit layer to disentangle the state. It is inexpensive, but a fixed-depth circuit cannot be improved by this method alone.
A. Analytic Decomposition by Disentangling
- Repeated disentangling layers form a circuit intended to approximate the original MPS, with the analytical procedure aimed at maximizing fidelity.A χ = 2 MPS maps exactly to one layer of two-qubit unitaries.Source: A. Analytic Decomposition by… · p. 4
- Each iteration truncates a copy of the MPS to χ = 2 by SVD, converts it to a two-qubit layer, then applies the layer’s inverse to the original MPS.The reduced-entanglement MPS is used for the next iteration, adding layers from the MPS backwards.Source: A. Analytic Decomposition by… · p. 4
- For fixed K, the analytic method cannot improve decomposition quality; high fidelity can require orders-of-magnitude more layers than the exact-decomposition lower bound K_min ∼ log2(χ_max).The passage recommends combining the method with constrained optimization of the circuit unitaries.Source: A. Analytic Decomposition by… · p. 4
B. Decomposition by Optimization
The optimization method improves a circuit approximation by updating its two-qubit unitaries one at a time. It derives each update from an environment tensor and constrains the result to remain unitary.
B. Decomposition by Optimization
- The algorithm iterates over circuit unitaries in sweeps, stopping at a preset sweep limit or when the target fidelity is reached.The circuit uses two-qubit U(4) gates in a linear nearest-neighbor topology.Source: B. Decomposition by Optimization · pp. 4–5
- For each circuit unitary, the method contracts the remaining tensor network to form a 4 × 4 environment tensor, then uses its SVD to obtain a locally optimal unitary update.The unitary component UV† minimizes Frobenius distance to the environment tensor and maximizes fidelity among unitary choices.Source: B. Decomposition by Optimization · pp. 5–6
- For the strong locally optimal update that increases fidelity with the MPS, a learning rate r ∈ [0, 1] influences the unitary update rule; optimization gave excellent performance at r = 0.6.Source: B. Decomposition by Optimization · p. 6
- Consecutive forward sweeps allow intermediate states to be cached and reused, avoiding full recalculation when constructing environment tensors.The work restricts its implementation to forward sweeps.Source: B. Decomposition by Optimization · p. 6
- Random initial circuits can increase the MPS bond dimension, so the combined approach uses analytic layers as initial guesses before optimizing fixed-depth circuits.The analytic layers tend to reduce bond dimension, while optimization seeks a better approximation without increasing circuit depth.Source: Figure 2 · p. 5, B. Decomposition by Optimization · p. 6
C. Combinations of analytic decomposition and optimization
The protocols combine analytic decomposition, identity-layer insertion, and constrained optimization in different ways. Iter[DiOall] grows the circuit sequentially using analytic steps and optimizes all layers created so far.
C. Combinations of analytic decomposition and optimization
- Analytic decomposition is relatively inexpensive but can require many layers, while brute-force optimization from random gates can require many sweeps and increase MPS bond dimension.Combining the methods is intended to balance classical resources against circuit depth.Source: C. Combinations of analytic… · p. 6
- The six protocols vary whether they use analytic decomposition or identity layers, and whether optimization targets the newest layer or all layers so far.The “Iter” protocols alternate layer creation with optimization; Dall and Oall use only analytic decomposition and optimization, respectively.Source: C. Combinations of analytic… · pp. 6–7
- Iter[DiOall] adds each new layer through one analytic decomposition step, then optimizes all existing layers; earlier layers are also optimized, unlike in Iter[DiOi].Both sequential protocols maximize the overlap at each decomposition step, but Iter[DiOi] keeps earlier layers fixed.Source: C. Combinations of analytic… · p. 7
- Iter[DiOall] is the most sophisticated combination; the other protocols serve as benchmarks that remove analytic decomposition, optimization, or sequential growth.The compared approaches correspond to Dall in Ref., Oall similar to Ref., and DallOall similar to Ref. [30].Source: C. Combinations of analytic… · p. 7
D. Discussion of Computational Complexity
The complexity discussion gives an overview rather than a rigorous resource derivation, because the protocols allow approximation error. It relates costs to MPS size, optimization sweeps, circuit depth, and bond dimension.
D. Discussion of Computational Complexity
- The complexity estimates are an overview, not a rigorous derivation, because the protocols permit approximation error rather than requiring exact MPS representation.The stated goal is to summarize expected complexities for the algorithms and protocols.Source: D. Discussion of Computational… · p. 7
- MPS memory requirements scale as ∼Nχ_max^2, while the stated MPS computational cost includes bringing the state into canonical form.The passage also notes that the MPS computational cost includes SVD on all bonds.Source: D. Discussion of Computational… · p. 7
- Sequential protocols incur an additional scaling factor of K because earlier, smaller layers are revisited during optimization.For exact decomposition of a generic MPS, the cited lower bound is K_min ∼ log2(χ_max), but the passage frames it as a best-case depth rather than a practical target.Source: D. Discussion of Computational… · p. 7
IV. EMPIRICAL ASSESSMENT
Iter[DiOall] is the most consistent and best-performing protocol across the tested MPS and optimization budgets, with its strongest gains at low budgets. The assessment compares six protocols on three 12-qubit, bond-dimension-64 MPS targets.
IV. EMPIRICAL ASSESSMENT
- The six protocols are tested on three 12-qubit, χmax = 64 MPS: a Heisenberg ground state, a 6 × 2 bars-and-stripes superposition, and a random MPS.The BAS middle-bond spectrum has degenerate large values or zeros; the random MPS has positive and negative entries and slower singular-value decay.Source: §IV · pp. 8–9
- Figure 4 reports infidelity (1 − f) versus circuit layers K at T = 10, 100, or 1000 sweeps; the low settings represent limited budgets, and no singular values are truncated.T = 1000 highlights robustness to local minima; contractions use the exact statevector.Source: §IV · p. 9
- Iter[DiOall] performs best and most consistently across the target MPS and sweep budgets; its largest low-budget gains reduce classical compute per target fidelity.The improvements are most significant at T = 10 and T = 100.Source: §IV · p. 9
- Dall and Iter[OiDi] improve slowly with depth; Oall needs many sweeps and can encounter local minima, while DallOall improves at low T but also faces minima.Iter[IiOall] is competitive on the ground-state and random MPS but underperforms on BAS, where Iter[DiOall] achieves significantly better fidelities.Source: §IV · p. 9
- Removing the analytical decomposition, circuit optimization, or sequential circuit growth makes Iter[DiOall] perform significantly worse.Source: §IV · p. 9
V. CONCLUSION
The paper concludes that Iter[DiOall] reliably decomposes MPS into high-fidelity shallow circuits, especially with few optimization steps, while noting that further quantitative studies are needed. It identifies extensions to tree tensor networks and possible quantum-hardware subroutines as follow-up directions.
V. CONCLUSION
- Across three 12-qubit, χ = 64 benchmarks, Iter[DiOall] most reliably decomposes MPS into high-fidelity shallow circuits, sometimes improving infidelity by orders of magnitude with limited optimization steps.It sequentially grows the circuit using analytical decomposition while interleaving optimization of existing circuit unitaries.Source: §V · p. 9
- Further quantitative studies are needed to evaluate the protocol’s performance and scalability.Source: §V · p. 9
- Extending the protocols to TTNs is proposed; PEPS conversion may be more challenging because strong canonical forms for exact conversion to multi-qubit-unitary circuits are lacking.The authors also suggest quantum-hardware subroutines for some tasks, including PEPS or layouts not classically efficient to simulate.Source: §V · pp. 9–10
Appendix A: Unitaries & Layers - Notation and Convention
The appendix defines notation for two-qubit unitaries and linear layers, and specifies conventions for ordering their application to states.
Appendix A: Unitaries & Layers - Notation and Convention
- U denotes a general U(4) unitary acting on two qubits; collections of unitaries are indexed by m.Source: Appendix A · p. 11
- L[U] denotes linear unitary layers in a staircase or next-neighbor topology; collections of layers are indexed by k.Source: Appendix A · p. 11
- Unitary order is indexed by its application to the ket; for layers in Sec. III A, order is reversed and each unitary is conjugated and transposed.Source: Appendix A · p. 11