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A Block Tensor Train Burer-Monteiro Framework for Low-Rank Quantum State Tomography
Shakir Showkat Sofi, Charlotte Vermeylen, Fatemeh Mohammadi, Lieven De Lathauwer
TL;DR
Quantum state tomography must recover physically valid density matrices despite exponential scaling with system size. The paper introduces a Block-TT Burer–Monteiro parameterization with DMRG optimization for compressed measurements, reporting accurate reconstruction and improved efficiency, while remaining targeted at low-mixedness states and requiring scaling of iterations and measurements for larger systems.
Problem
QST faces exponential measurement and computational growth for generic states while requiring Hermiticity, positive semidefiniteness, and unit trace.
Method
The framework represents density matrices using Block-TT factors and develops single-site and two-site DMRG algorithms with efficient tensor-network contractions and adaptive rank refinement.
Results
DMRG-I and DMRG-II achieve fidelity ≥0.99 for α ≥0.5, lower trace-distance error, and over an order-of-magnitude faster runtime than CVX and BM.
Takeaways & Limitations
The framework provides a physically valid, memory- and computation-efficient approach for learning low-rank mixed quantum states from compressed measurements.
Takeaways & Limitations
The parameterization is efficient only when mixedness is low, and larger systems require iterations and measurement counts to scale strongly with N.
Abstract
from arXiv · showhide
Quantum state tomography is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.
1. Introduction.
Quantum state tomography reconstructs quantum states from measurements, but generic reconstruction faces exponential scaling and physicality constraints. The paper addresses this challenge with a Block-TT framework and DMRG algorithms for compressed-sensing tomography.
- Motivation: QST reconstructs a quantum state from measurement samples, while generic measurement and computational requirements grow exponentially with the number of qubits.Reconstructed states must also satisfy Hermiticity, positive semidefiniteness, and unit-trace constraints.
- Related approaches: Low-rank and tensor-network methods exploit structure in states such as ground states, but existing factorized models can retain exponential parameter dependence.Tensor-network approaches include MPS/TT, tree tensor networks, and projected entangled pair states.
- Contributions: The paper presents a Block-TT Burer–Monteiro framework that compresses the parameter space from exponential to linear in the number of qubits or qudits.The framework extends preliminary conference work and targets high-dimensional mixed states through compressed sensing.
- Contributions: The proposed methods include single-site and two-site DMRG algorithms operating directly on Block-TT factors with adaptive rank refinement.They use effective operators and recursively constructed environments for expectation evaluation and local optimization.
2. Preliminaries.
This section introduces tensor, matrix, block-matrix, and tensor-network notation used to formulate the proposed tomography algorithms. It also presents contraction and Kronecker-type operations for describing network structure.
- Notation: The paper distinguishes scalars, vectors, matrices, block matrices, and tensors through notation, and uses unfoldings and interface matrices to describe tensor-network components.Blockwise conjugate transpose notation is also introduced for block matrices.
- Tensor operations: Tensor-network contractions combine tensors over common modes and provide a multilinear extension of matrix–matrix multiplication.The C-product combines outer block indices through a Kronecker layout and inner blocks through matrix multiplication.
- Tensor operations: The strong Kronecker product combines block matrices using Kronecker products in place of ordinary matrix products.Tensor-network diagrams visualize these operations, including distinct block-layout and inner-matrix edges.
- Quantum preliminaries: A qubit is represented in a two-dimensional Hilbert space, with pure states expressed as normalized superpositions of basis states.The squared magnitudes of the amplitudes determine measurement probabilities in the chosen basis.
State representation.
Quantum states may be pure vectors or mixed-state density matrices, while measurements are represented by operators whose expectation values determine observed statistics. Informational completeness is required for unique state inference.
- State representation: Pure states of an N-qudit system are unit vectors in the tensor-product Hilbert space of the constituent qudits.The single-qudit pure-state space is described using vectors in C^d.
- State representation: Mixed states are probabilistic ensembles of pure states represented by density matrices, with state and density matrix used interchangeably.The ensemble assigns probabilities to constituent pure-state vectors.
- Measurements: Observable quantities are represented by Hermitian operators, and measurement probabilities are obtained from traces such as Tr(ρP_k).Projective measurements use orthogonal projectors satisfying completeness.
- Measurements: POVMs generalize projective measurements by requiring positivity and completeness without requiring orthogonality or projection.A POVM can contain more elements than the Hilbert-space dimension, with outcome probabilities Tr(ρE_k).
- Tomography: Pauli observables and SIC POVMs provide measurement constructions used for single- and multi-qubit tomography.Local multi-qubit Pauli observables are formed as Kronecker products of single-qubit Pauli matrices.
- Tomography: QST uses measurement statistics from informationally complete POVMs or combined POVMs to infer the density matrix.For N-qudit systems, the Hilbert-space dimension is D=d^N.
Low-rank QST.
Low-rank QST reduces measurement requirements by exploiting density-matrix structure, but conventional low-rank factorization can still scale exponentially with system size. Tensor trains provide a route toward compressed parameterizations.
- Low-rank QST: Generic density-matrix recovery requires O(D^2) measurement settings, whereas rank-R states can require O(RD log^2 D) Pauli measurements or O(RD) structured measurements.These bounds concern unique estimation under the stated measurement models.
- Rank minimization: Nuclear-norm methods enforce low rank implicitly through a convex relaxation subject to measurement-fitting and PSD constraints.The solution may be normalized to unit trace, while direct rank minimization is non-convex and NP-hard.
- Error minimization: Burer–Monteiro methods explicitly parameterize the density matrix with fixed-rank factors, improving computational efficiency while making optimization non-convex.Hermitian and PSD constraints are automatically satisfied by construction, although local solutions may occur.
- Block-TT motivation: The Block-TT extension breaks the exponential parameter dependence of ordinary Burer–Monteiro factors by representing the factors in Block-TT format.The cited contribution explicitly contrasts exponential dependence in A with the proposed Block-TT representation.
- Tensor trains: Tensor trains represent high-order tensors through contracted TT-cores whose rank tuple controls the representation complexity.TT-SVD constructs cores sequentially using truncated singular-value decompositions.
- Tensor trains: Tensorization reshapes large vectors and matrices into tensors with compact TT representations and structured cores.Matrix TT cores split each long mode into separate row and column modes.
TT representations for vectors and matrices.
Tensor-train representations factor high-order vectors and matrices into chains of low-order cores. Block-TT extends this structure by allowing one core to carry multiple shared vector components, supporting frame-based and DMRG-style operations.
- TT-cores encode tensorized vectors or matrices, while TT-ranks capture the internal structural complexity of the representation.
- A matrix TT represents a tensorized matrix as a sequential strong Kronecker product of N block matrices.
- For vectors, setting J=1 reduces each block to a vector and recovers the standard TT/MPS representation.
- A Block-n-TT has vector-valued blocks except at one matrix-valued core and is equivalent to K TT vectors sharing all other cores.
- Block-TT addition and multiplication combine cores through direct sums and C-products, respectively, while TT rounding controls the resulting ranks.
- The Block-TT representation is linear in its block-bearing core, enabling a matrix frame equation and corresponding one- or two-site formulations.
- Orthogonalizing TT cores with QR or LQ decompositions produces n-orthogonal forms and column-wise orthogonal frame matrices.
- The framework uses a contraction of two Block-TT networks to enforce Hermiticity and positivity and enable DMRG-like optimization for low-rank QST.
3. Block-TT approach to low-rank QST.
The Block-TT approach factors a low-rank density matrix through a tall Block-TT factor and its Hermitian transpose. The block index bounds the represented matrix rank while retaining shared tensor-network structure.
- The density matrix is represented as ρTT = ATTAH_TT, where ATT is a Block-TT factor with D=d^N rows and K columns.
- The factor ATT bounds the rank of ρTT by K and uses vector-valued cores except for one matrix-valued core carrying the block index.
- The Block-TT factor can equivalently be viewed as K TT vectors that share every core except the block-bearing core.
Physicality constraints.
Under n-orthogonality, physicality properties of the global density matrix are represented locally through an effective density matrix. The resulting low-rank formulation supports compressed measurements and efficient tensor-network optimization.
- Under n-orthogonality, the effective density matrix shares the nonzero eigenvalues of the global density matrix.
- The Gram structure of the effective density matrix guarantees that ρTT is positive semidefinite by construction.
- The global density matrix’s structural properties are reflected locally, allowing suitable SDP constraints to reduce to an effective problem under frame projections.
- The optimization operates on low-rank factors rather than directly on the effective density matrix, yielding a nonconvex local problem whose orthogonality center can be shifted across cores.
- The Block-TT parameterization is efficient only for low-mixedness states and uses dR^2(log_d D −3) + dR^2K + 2dR parameters when R_n=R.
- Compressed trace-based measurements return observable expectation values that generally depend on all density-matrix entries.
- Low-rank TTM measurement operators enable efficient expectation-value contractions, followed by single-site DMRG optimization and a two-site extension.
DMRG-based approach to scalable QST.
The approach combines Block-TT parameterization with DMRG optimization for compressed-sensing tomography, using effective operators and recursive environments to evaluate measurements efficiently. Single-site and two-site sweeps optimize local cores while shifting the block index and adapting TT ranks.
- Efficient expectation-value evaluation: Expectation values are evaluated through core-wise contractions between the Block-TT state representation and tensorized measurement operators.The equivalent effective-operator form is constructed from tensor-network contractions rather than by explicitly forming the full operator.
- Efficient expectation-value evaluation: Effective operators are built from recursively computed left and right environments with boundary conditions at both ends of the chain.The environments are updated by contracting neighboring environments, cores, and measurement-operator cores in an ordered sequence.
- Single-site DMRG: Single-site DMRG converts the global problem into linked local optimization problems under an orthogonality condition.Projected gradient descent is one stated option for solving each constrained local subproblem before moving the block index and orthogonality center.
- Rank adaptation: During single-site sweeps, truncated SVD updates TT ranks while shifting the block index and orthogonality center between adjacent cores.For K = 1, however, the single-site update cannot increase the TT-rank.
- Two-site DMRG: Two-site DMRG optimizes adjacent cores jointly, then splits the combined core by truncated SVD to enable rank adaptation and faster convergence per iteration.The larger local tensor increases memory and computational requirements, and the resulting local problems can slow optimization per iteration.
- Local measurements: When measurement operators are identity outside a local subsystem, the full-state trace reduces to an equivalent trace on the reduced density matrix.This follows from tracing out the fixed subsystems.
4. Numerical experiments.
The experiments evaluate accuracy, convergence, rank adaptivity, scalability, and large-scale tomography using compressed local measurements. The proposed DMRG methods generally reconstruct accurately and efficiently, while performance depends on sampling, system size, model capacity, and measurement overlap.
- 4.1. Accuracy and computational efficiency.: The proposed DMRG methods outperform CVX and Burer–Monteiro baselines in accuracy and efficiency, achieving near-perfect fidelity for α ≥0.5 with over an order-of-magnitude speedup.Increasing α improves reconstruction accuracy with minor computational cost; DMRG-I performs slightly better at α = 0.25, while DMRG-II becomes superior as α increases.
- 4.2. Rank adaptivity and scalability.: DMRG-II converges within one sweep and reaches Rmax = 10 by sweep 1, whereas DMRG-I adapts more gradually and achieves similar convergence in two sweeps.The comparison uses an N = 7 state with true Rmax = 10 and records median loss over 10 trials.
- 4.2. Rank adaptivity and scalability.: For N ≤6, DMRG-II is more accurate and faster, but its accuracy advantage disappears and runtime grows faster than DMRG-I for larger systems.The slowdown is attributed to faster TT-rank growth, more expensive two-site core solves, and SVDs.
- 4.2. Rank adaptivity and scalability.: As system size increases, fixed sweep budgets cause reconstruction error to rise, so both iteration counts and measurement counts must scale strongly with N.This limitation is reported alongside the scalability comparison from N = 4 to N = 12.
- 4.2. Rank adaptivity and scalability.: Reconstruction fidelity increases substantially when model capacity reaches Ktrue, while capacities above the true value slightly reduce fidelity by fitting measurement noise.Smaller K values require longer optimization times, whereas larger values converge faster; K = 2 achieves the best fidelity in the reported experiment.
- 4.3. Exploiting overlapping measurements for large-scale QST.: Overlapping localized measurements support tomography of an N = 30 transverse-field Ising ground state represented as an MPS/TT with Rmax = 6.Window width and stride trade reconstruction fidelity against computational cost, but adequate overlap and total measurement budget are required to capture relevant correlations.