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arXiv:2610.02103

Unitary Schur Sampling of Qudits via Random SWAP Tests: Hunt for Antisymmetry

Shrigyan Brahmachari, David Jakab, Henry D. Pfister, Iman Marvian

quant-phcond-mat.stat-mechhep-thmath-phnucl-thSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

Schur sampling is a fundamental primitive for extracting permutation-invariant information from many-body quantum systems and has broad applications in quantum information science. Circuit-based implementations typically rely on coherent representation-theoretic operations such as Clebsch-Gordan transforms, generalized phase estimation, or quantum Fourier transforms over the symmetric group. We show that unitary Schur sampling of arbitrary permutation-invariant mixed states on $n$ $d$-dimensional qudits can instead be implemented using only pairwise SWAP tests, namely, two-qudit projective measurements onto the symmetric and antisymmetric subspaces. Our protocol achieves error $ε$ in diamond distance using $O(n\min\{n,d\}^{3}\log^2(n/ε))$ random SWAP tests. The central idea is to repeatedly identify and extract the largest antisymmetric subsystem. This turns unitary Schur sampling into a search for antisymmetry and gives the algorithm a natural interpretation as a stochastic traversal of a Young diagram. The protocol preserves the $\mathrm{SU}(d)$-irrep state while preparing a canonical pure state in the multiplicity subsystem. As an application, we consider quantum purity amplification (QPA), in which multiple noisy copies of a pure quantum state are combined to produce a state of higher purity. We show that the optimal purified state can be obtained by retaining a single designated output qudit and discarding the rest.

Comments
19 pages + 29 pages of Appendices, 2 Figures, Preliminary version

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Schur sampling is a fundamental primitive for extracting permutation-invariant information from many-body quantum systems and has broad applications in quantum information science. Circuit-based implementations typically rely on coherent representation-theoretic operations such as Clebsch-Gordan transforms, generalized phase estimation, or quantum Fourier transforms over the symmetric group. We show that unitary Schur sampling of arbitrary permutation-invariant mixed states on nnn ddd-dimensional qudits can instead be implemented using only pairwise SWAP tests, namely, two-qudit projective measurements onto the symmetric and antisymmetric subspaces. Our protocol achieves error εεε in diamond distance using O(nmin⁡{n,d}3log⁡2(n/ε))O(n\min\{n,d\}^{3}\log^2(n/ε))O(nmin{n,d}3log2(n/ε)) random SWAP tests. The central idea is to repeatedly identify and extract the largest antisymmetric subsystem. This turns unitary Schur sampling into a search for antisymmetry and gives the algorithm a natural interpretation as a stochastic traversal of a Young diagram. The protocol preserves the SU(d)\mathrm{SU}(d)SU(d)-irrep state while preparing a canonical pure state in the multiplicity subsystem. As an application, we consider quantum purity amplification (QPA), in which multiple noisy copies of a pure quantum state are combined to produce a state of higher purity. We show that the optimal purified state can be obtained by retaining a single designated output qudit and discarding the rest.
We formalize the task of unitary Schur sampling -- an extension of weak Schur sampling -- which is the process of measuring the Young label and the unitary group register of an input mmm qudit state. Intuitively, this task is equivalent to applying the Schur transform, projecting onto the isotypic subspaces of the unitary and symmetric groups indexed by the Young labels, and discarding of the permutation register. As such unitary Schur sampling is the natural task in processes such as quantum state tomography or spectrum estimation. We generalize this task to unitary mixed Schur sampling to account for the recently introduced mixed Schur-Weyl transform. We provide a streaming algorithm which achieves an exponential reduction in the memory complexity and a polynomial reduction in the gate complexity over naïve algorithms for the task of unitary (mixed) Schur sampling. Further, we show that if the input state has limited rank, the gate and memory complexities of our streaming algorithm as well as the algorithms for the full Schur and mixed Schur transforms are further reduced. Our work generalizes and improves on the results in arXiv2309.11947.
We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-rrr qudit state ρ=∑i=1rλi∣φi⟩ ⁣⟨φi∣ρ=\sum_{i=1}^r λ_i |\varphi_i\rangle\!\langle\varphi_i|ρ=∑i=1r​λi​∣φi​⟩⟨φi​∣, prepare a copy of the state proportional to Πantir(∣φ1⟩⊗…⊗∣φr⟩)Π_{\mathrm{anti}}^r (|\varphi_1\rangle\otimes \ldots\otimes|\varphi_r\rangle)Πantir​(∣φ1​⟩⊗…⊗∣φr​⟩), with ΠantirΠ_{\mathrm{anti}}^rΠantir​ the anti-symmetric subspace projector. We first give a simple algorithm for anti-symmetrization that uses O~(r/λmin⁡)\widetilde{O}(r / λ_{\min})O(r/λmin​) copies of ρρρ, where λmin⁡λ_{\min}λmin​ is the minimum non-zero eigenvalue of ρρρ. This algorithm works even in the streaming setting with small working memory, and it uses only simple quantum operations like controlled-SWAPs and constantly many layers of single-qubit gates. As an application of this streaming anti-symmetrization procedure, we give a reduction from gapped mixed-state one-way state generators to pure-state one-way state generators. Then, using tools from Schur-Weyl duality, we give and analyze a second (non-streaming) anti-symmetrization procedure that uses O(r/λmin⁡)O(r / λ_{\min})O(r/λmin​) copies of ρρρ, which then serves as a building block in new algorithms for weak Schur sampling and unitary Schur sampling, outperforming existing approaches in some regimes of number of copies nnn and local dimension ddd, as well as in an algorithm for optimal purity amplification for qudit states.
The quantum Schur transform maps the computational basis of a system of nnn qudits onto a \textit{Schur basis}, which spans the minimal invariant subspaces of the representations of the unitary and the symmetric groups acting on the state space of nnn ddd-level systems. We introduce a new algorithm for the task of weak Schur sampling. Our algorithm efficiently determines both the Young label which indexes the irreducible representations and the multiplicity label of the symmetric group. There are two major advantages of our algorithm for weak Schur sampling when compared to existing approaches which proceed via quantum Schur transform algorithm or Generalized Phase Estimation algorithm. First, our algorihtm is suitable for streaming applications and second it is exponentially more efficient in its memory usage. We show that an instance of our weak Schur sampling algorithm on nnn qubits to accuracy εεε requires only O(log⁡2n)O(\log_2n)O(log2​n) qubits of memory and O(n3log⁡2(nε))O(n^3\log_2(\frac{n}ε))O(n3log2​(εn​)) gates from the Clifford+T set. Further, we show that our weak Schur sampling algorithm on nnn qudits decomposes into O(dn2dlog⁡2p(n2dε))O\big(dn^{2d}\log_2^p\big(\frac{n^{2d}}ε\big)\big)O(dn2dlog2p​(εn2d​)) gates from an arbitrary fault-tolerant qudit universal set, for p≈4p\approx 4p≈4, and requires a memory of O(log⁡dn)O(\log_dn)O(logd​n) qudits to implement.
Tight bounds on quantum sample complexity and quantum query complexity have been known for various computational problems in the literature, whereas tight bounds on quantum time complexity (i.e., the size of quantum circuits) remain unresolved. In this paper, we provide a framework to establish lower bounds on the quantum time complexity for testing permutation-invariant properties of quantum states, via a reduction from quantum sample complexity. As an application, we obtain a series of matching lower bounds when given sample access to the input quantum states, including: 1. The SWAP test due to Buhrman, Cleve, Watrous, and de Wolf (Phys. Rev. Lett. 2001) is time-optimal to estimate the purity tr⁡(ρ2)\operatorname{tr}(ρ^2)tr(ρ2) and the inner product tr⁡(ρσ)\operatorname{tr}(ρσ)tr(ρσ). 2. The Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki, and Kwek (Phys. Rev. Lett. 2002) is time-optimal to estimate the high-order functionals tr⁡(ρk)\operatorname{tr}(ρ^k)tr(ρk). 3. The productness tester for multipartite pure states due to Harrow and Montanaro (J. ACM 2013) is time-optimal. 4. The LMR protocol due to Lloyd, Mohseni, and Rebentrost (Nat. Phys. 2014) is time-optimal to implement the reflection operator about a pure state. 5. The samplizer due to Wang and Zhang (IEEE Trans. Inf. Theory 2025) is time-optimal for pure states. 6. The estimator for pure-state trace distance and fidelity due to Wang and Zhang (ICALP 2026) is time-optimal. To the best of our knowledge, this is the first method that allows us to systematically establish tight lower bounds on quantum time complexity.
Many quantum information tasks use inputs of the form ρ⊗mρ^{\otimes m}ρ⊗m, which naturally induce permutation and unitary symmetries. We classify all quantum channels that respect both symmetries - i.e. unitary-equivariant and permutation-invariant quantum channels from (Cd)⊗m(\mathbb{C}^{d})^{\otimes m}(Cd)⊗m to (Cd)⊗n(\mathbb{C}^{d})^{\otimes n}(Cd)⊗n - via their extremal points. Operationally, each extremal quantum channel factors as unitary Schur sampling →\rightarrow→ an irrep-level unitary-equivariant quantum channel →\rightarrow→ the adjoint unitary Schur sampling. We give a streaming implementation ansatz that uses an efficient streaming implementation of unitary Schur sampling together with a resource-state primitive, and we apply it to state symmetrization, symmetric cloning, and purity amplification. In these applications we obtain polynomial-time algorithms with exponential memory improvements in m,nm,nm,n. Further, for symmetric cloning we present, to our knowledge, the first efficient (polynomial-time) algorithm with explicit memory and gate bounds.
We study the following quantum state identity problem: given access to copies of nnn unknown pure quantum states, the goal is to determine whether every pair has fidelity at least 1−ε1-\varepsilon1−ε or some pair has fidelity at most ε\varepsilonε. This problem relaxes the identical-or-orthogonal promise considered in previous work. For any fixed overlap parameter ε∈(0,1/4)\varepsilon\in(0,1/4)ε∈(0,1/4), we present two quantum algorithms with complementary guarantees on circuit depth and sample complexity. The first runs in constant depth using parallel swap tests and takes O(log⁡(n/δ)log⁡(2/δ))O(\log(n/δ)\log(2/δ))O(log(n/δ)log(2/δ)) copies of each input state to succeed with probability at least 1−δ1-δ1−δ. The second uses the Schur transform to reduce the sample complexity to O(log⁡(n/δ))O(\log(n/δ))O(log(n/δ)), through an analysis relating Jucys--Murphy elements to Schur sampling. We also prove a matching sample complexity lower bound of Ω(log⁡(n/δ))Ω(\log(n/δ))Ω(log(n/δ)) for any quantum algorithm with failure probability at most δ≤1/2−cδ\le 1/2-cδ≤1/2−c, for any constant c>0c>0c>0, establishing that the second algorithm achieves the optimal dependence on both nnn and δδδ for fixed ε\varepsilonε.