random-dimension-reduction-quantum-learning

name	random-dimension-reduction-quantum-learning
description	Random dimension reduction procedure for quantum states that reduces dimensions while preserving properties invariant under tensor power action of isometries. Provides black-box method to replace dimension with max rank in sample complexity for learning symmetric properties, including multi-state estimation of distances, fidelities, and relative entropies.
metadata	{"arxiv_id":"2606.23592","published":"2026-06-22","authors":"Angus Lowe, Xinyu Tan","tags":["quantum-learning","dimension-reduction","state-tomography","symmetric-properties","sample-complexity"]}

Random Dimension Reduction for Quantum States

Core Concepts

A procedure that simultaneously reduces the dimension of many quantum states while preserving properties invariant under the tensor power action of an isometry. This enables sample-optimal learning of symmetric properties with complexity depending on max rank rather than dimension.

Key Results

• Dimension → Rank: Sample complexity for symmetric properties depends on max rank of input states, not Hilbert space dimension
• Full tomography after reduction: yields improved upper bounds for estimating distances, fidelities, and relative entropies between state pairs
• Efficient circuit: implementable using the Schur transform
• Connection to random purification: Choi-Jamiolkowski representation reveals connection to Tang-Wright-Zhandry random purification channel

Methodology

Step 1: Random Dimension Reduction Procedure

For a collection of states {ρ₁, ..., ρₘ} on H_d:

1. Apply a random isometry V: H_d → H_r where r = max rank
2. Output reduced states {Vρ₁V†, ..., VρₘV†} on H_r
3. Symmetric properties are preserved under this mapping

Step 2: Sample Complexity Reduction

For learning any symmetric property f(ρ₁, ..., ρₘ):

• Before: O(d²/ε²) samples
• After reduction: O(r²/ε²) samples where r = max rank(ρᵢ)
• When r ≪ d, this is an exponential improvement

Step 3: Circuit Implementation

Implement via Schur transform:

1. Decompose (C^d)^{⊗n} using Schur-Weyl duality
2. Project onto irrep subspaces labeled by partitions
3. Extract reduced state on smaller representation space

Usage Patterns

Pattern 1: Low-Rank State Tomography

When target states are approximately low-rank (common in noisy quantum devices):

• Apply dimension reduction → full tomography on reduced space
• Achieves same accuracy with fewer samples

Pattern 2: Multi-State Property Estimation

For comparing multiple quantum states:

• Distance estimation: ||ρ - σ||₁
• Fidelity estimation: F(ρ, σ)
• Relative entropy: S(ρ||σ) All benefit from rank-dependent rather than dimension-dependent scaling

Pattern 3: Quantum Learning with Symmetry

Any learning task with permutational symmetry in the samples:

• State discrimination
• Property testing
• Quantum hypothesis testing

Pitfalls

• Only symmetric properties: The reduction preserves only properties invariant under U^{⊗n} action
• Rank estimation: Requires prior knowledge or estimation of max rank
• Schur transform cost: O(n² log² d) gates for n copies on d-dimensional space

Activation Keywords

• random dimension reduction quantum
• quantum state tomography sample complexity
• symmetric property estimation quantum
• Schur transform state learning
• quantum fidelity estimation rank
• quantum dimension reduction
• 量子态降维学习