Weak Schur Sampling

• Weak Schur sampling is a measurement technique in Schur–Weyl duality that outputs only the Young diagram label, discarding finer multiplicity details.
• It plays a crucial role in quantum inference tasks such as spectrum estimation, state tomography, entanglement concentration, and purification, with implementations including streaming algorithms and random SWAP tests.
• Recent advances feature efficient streaming methods using logarithmic quantum memory and refined unitary and mixed sampling protocols that improve resource efficiency.

Weak Schur sampling is the standard coarse Schur–Weyl measurement on (n) qudits: one applies a Schur transform and measures only the Young label (\lambda) indexing the isotypic component, while discarding finer multiplicity and internal irrep data. In the qubit case this label is equivalent to the total spin (j) or (J). The task sits strictly below full Schur-basis measurement, but it is already operationally significant in settings such as spectrum estimation, tomography, and related permutation-invariant inference tasks. Recent work has clarified both its formal status inside Schur–Weyl duality and several distinct implementation regimes: a streaming weak Schur sampler with logarithmic quantum memory, memory-efficient extensions that output the unitary-group register rather than the full Schur state, and a qubit-specific realization via random SWAP tests for permutation-invariant inputs [2309.11947] [2410.15793] [2508.05046].

1. Schur–Weyl setting and the weak measurement

On (n) qudits of local dimension (d), Schur–Weyl duality gives the decomposition
[
(\mathbb C^d)^{\otimes n}\cong \bigoplus_{\lambda\vdash_d n}\mathcal P_\lambda\otimes \mathcal Q_\lambda^d,
]
where (\lambda\vdash_d n) is a partition of (n) into at most (d) parts, (\mathcal P_\lambda) is the symmetric-group irrep space, and (\mathcal Q_\lambda^d) is the (U(d)) or (SU(d)) irrep space. The natural commuting actions are the permutation action (\mathbf P(\sigma)) on tensor factors and the diagonal unitary action (\mathbf Q(U)). A Schur transform (U_{\rm Sch}) is any unitary that maps the computational basis to a basis adapted to this direct-sum decomposition [2309.11947] [2410.15793].

Weak Schur sampling is the PVM that measures only which Schur–Weyl block is occupied. In the formulation emphasized in recent work, the projectors are
[
{\Pi_\lambda}{\lambda\vdash_d n}, \qquad \Pi\lambda=\mathbb I_{\mathcal P_\lambda}\otimes \mathbb I_{\mathcal Q_\lambda^d},
]
acting on the Schur-transformed space, and the outcome law for an input state (\rho) is
[
p(\lambda,\rho)=\operatorname{Tr}!\left[\big(U_{\rm Sch}\rho U_{\rm Sch}^\dagger\big)\Pi_\lambda\right].
]
Equivalently, in the notation of the streaming weak Schur sampling work,
[
\Pr[\lambda]=\operatorname{Tr}!\left[\rho\,\Pi_\lambda^{\mathrm{Std}}\right].
]
The post-measurement state remains supported on (\mathcal P_\lambda\otimes\mathcal Q_\lambda^d); weak Schur sampling measures only the block label and does not resolve basis states within that block [2410.15793] [2309.11947].

For qubits, the relevant partitions have at most two rows. The Young label is equivalent to total spin:
[
\lambda=(n/2+j,\; n/2-j).
]
Hence, in the (\mathrm{SU}(2)) setting, weak Schur sampling can be read either as measuring (\lambda) or as measuring the total angular momentum (j) [2508.05046].

2. Weak, strong, and unitary Schur sampling

A Schur basis vector is typically labeled by a triple
[
(\lambda,p_\lambda,q_\lambda),
]
where (\lambda) specifies the isotypic block, (p_\lambda) indexes a basis of the symmetric-group multiplicity space (\mathcal P_\lambda), and (q_\lambda) indexes a basis of the unitary-group irrep space (\mathcal Q_\lambda^d). In this standard terminology, weak Schur sampling outputs only (\lambda), whereas strong Schur sampling resolves the full Schur-basis label ((\lambda,p_\lambda,q_\lambda)) [2309.11947].

The 2024 work formalizes a task it calls unitary Schur sampling, explicitly as an extension of weak Schur sampling. If weak Schur sampling on (\rho) yields label (\lambda) with probability (p(\lambda,\rho)) and post-measurement state (\rho_\lambda) on (\mathcal P_\lambda\otimes\mathcal Q_\lambda^d), then unitary Schur sampling returns
[
\lambda
\quad\text{and}\quad
\operatorname{Tr}{\mathcal P\lambda}[\rho_\lambda]
]
with probability (p(\lambda,\rho)). Operationally, this is “apply the Schur transform; project onto the isotypic subspaces indexed by (\lambda); discard the permutation register.” The point is not that more is measured than in weak Schur sampling, but that the retained post-measurement object is the reduced (SU(d))-irrep state rather than the full Schur state [2410.15793].

This distinction matters because several applications cited in that work depend only on the unitary-group register: spectrum estimation, quantum state tomography, entanglement concentration, purification, optimal cloning, and quantum majority vote. In the qubit permutation-invariant setting, the random-SWAP approach expresses the same hierarchy in (\mathrm{SU}(2)) language: weak Schur sampling means outputting only (j), while unitary Schur sampling means outputting (j) together with the corresponding sector state (\sigma_j) [2410.15793] [2508.05046].

3. Streaming weak Schur sampling with logarithmic quantum memory

A major algorithmic development is the streaming weak Schur sampling algorithm of 2023. Its central idea is to avoid the full Schur transform altogether. Instead of coherently maintaining all Schur branches, the algorithm processes qudits one at a time, applies only the single Clebsch–Gordan update relevant to the currently occupied irrep, and immediately measures the next Young label. After the (k)-th qudit has been processed, the working quantum state lies in one specific (U(d))-irrep (\mathcal Q_\lambda^d) for some (\lambda\vdash k); when the ((k+1))-st qudit arrives, one applies
[
U_\lambda^{\mathrm{CG}}:\mathcal Q_\lambda^d\otimes \mathcal Q_{(1)}^d \to \bigoplus_{j=0}^{d-1}\mathcal Q_{\lambda+\mathbf e_j}^d,
]
and measures which child partition (\lambda+\mathbf e_j) occurred [2309.11947].

Because the computation collapses back to a single irrep after each step, the quantum memory stores only the current irrep state and the current Young label, not the full (n)-qudit input or a coherent superposition over all Schur blocks. The resulting correctness statement is exact:
[
\Pr[\Lambda^{\mathrm{wSch}}=\lambda]=\operatorname{Tr}!\left[\rho\,\Pi_\lambda^{\mathrm{Std}}\right].
]
The same work emphasizes that the observed path
[
(1)=\lambda^1\to \lambda^2\to \cdots \to \lambda^n=\lambda
]
also determines a multiplicity label (p_\lambda\in\mathcal P_\lambda). In that sense the algorithm is slightly stronger than minimal weak Schur sampling: it directly outputs (\lambda), and the path recovers the symmetric-group multiplicity label, but it does not output the full unitary irrep basis label (q_\lambda) [2309.11947].

The resource bounds are correspondingly sharp. For (n) qubits, an implementation to accuracy (\epsilon) requires only (O(\log_2 n)) qubits of memory and
[
O!\left(n^3\log_2!\left(\frac{n}{\epsilon}\right)\right)
]
gates from the Clifford+(T) set. For (n) qudits, the stated bounds are (O(\log_d n)) qudits of memory and
[
O!\left(d\,n^{2d}\log_2^p!\left(\frac{n^{2d}}{\epsilon}\right)\right),
\qquad p\approx 4,
]
over an arbitrary fault-tolerant universal qudit gate set. The paper contrasts this with implementations via the full Schur transform or generalized phase estimation, which require (O(n)) quantum memory and are not naturally streaming in the same way [2309.11947].

4. Unitary and mixed Schur sampling as refinements of the weak task

The 2024 extension reframes weak Schur sampling inside a broader operational hierarchy. In the ordinary Schur–Weyl setting on (m) qudits,
[
(\mathbb C^d)^{\otimes m}\cong \bigoplus_{\lambda\vdash_d m}\mathcal P_\lambda\otimes \mathcal Q_\lambda^d,
]
unitary Schur sampling returns (\lambda) and the reduced state on (\mathcal Q_\lambda^d). The same paper then passes to mixed Schur–Weyl duality on
[
(\mathbb C^d)^{\otimes m}\otimes (\overline{\mathbb C^d})^{\otimes n},
]
where the decomposition is indexed not by partitions (\lambda) but by staircases (\gamma):
[
(\mathbb C^d)^{\otimes m}\otimes (\overline{\mathbb C^d})^{\otimes n}
\cong
\bigoplus_{\gamma\vdash_d(m,n)} \mathcal P_\gamma^d \otimes \mathcal Q_\gamma^d.
]
Here the multiplicity side is associated with the walled Brauer algebra rather than (\mathbb C[\mathcal S_m]) [2410.15793].

The corresponding task, unitary mixed Schur sampling, takes an input (\rho), measures (\gamma), and outputs
[
\operatorname{Tr}{\mathcal P\gamma^d}[\rho_\gamma]
]
with probability (p(\gamma,\rho)). The streaming algorithm again processes one tensor factor at a time. If the current irrep label is (\gamma^k), the algorithm applies a Clebsch–Gordan transform (U_{\rm CG}^{\gamma^k}) when the next factor is (\mathbb C^d), or a dual Clebsch–Gordan transform (U_{\rm dCG}^{\gamma^k}) when the next factor is (\overline{\mathbb C^d}), then measures the next admissible irrep. Multiplicity information appears classically as the observed path (p_\gamma=(\gamma^1,\dots,\gamma^{m+n})), so the full multiplicity register never has to be stored coherently [2410.15793].

The correctness statement matches the Schur or mixed-Schur PVM exactly:
[

\Pr[\Gamma=\gamma]

\operatorname{Tr}!\left[
\big(U_{\rm Sch}^{m,n}\rho (U_{\rm Sch}^{m,n})^\dagger\big)\Pi_\gamma^{m,n}
\right].
]
For ordinary Schur sampling, one sets (n=0). The reported resource bounds for unitary mixed Schur sampling to accuracy (\epsilon) are
[
M=O\big(d^2\log_2^{p}(d,m,n,1/\epsilon)\big), \qquad
T=O\big((m+n)d^4\log_2^{p}(d,m,n,1/\epsilon)\big),
]
with (p\approx 1.44). In the ordinary case (n=0), this becomes
[
M=O\big(d^2\log_2^{p}(d,m,1/\epsilon)\big), \qquad
T=O\big(md^4\log_2^{p}(d,m,1/\epsilon)\big).
]
Under reduced-rank promises, the same paper gives the improved bounds
[
M=O\big((r+r')d\log_2^{p}(d,m,n,1/\epsilon)\big), \qquad
T=O\big((m+n)(r+r')^3d\log_2^{p}(d,m,n,1/\epsilon)\big),
]
and for ordinary Schur sampling,
[
M=O\big(rd\log_2^{p}(d,m,1/\epsilon)\big), \qquad
T=O\big(mr^3d\log_2^{p}(d,m,1/\epsilon)\big).
]
This generalizes and improves on the 2023 weak Schur sampling result [2410.15793].

5. Qubit weak Schur sampling via random SWAP tests

A distinct implementation paradigm appears in the 2025 work on random SWAP tests. There the setting is (n) qubits and permutation-invariant states. The (\mathrm{SU}(2))–(\mathbb S_n) decomposition is written as
[
(\mathbb C^2)^{\otimes n}
\cong
\bigoplus_{j=j_{\min}}^{n/2}
\left(\mathbb C^{2j+1}\otimes \mathbb C^{m(n,j)}\right),
]
where
[
m(n,j)=\binom{n}{\frac n2-j}\frac{2j+1}{\frac n2+j+1}.
]
For a permutation-invariant state (\sigma),
[
\sigma=
\bigoplus_j
p_j\left(\sigma_j\otimes \frac{\mathbb I_{m(n,j)}}{m(n,j)}\right).
]
In this notation, weak Schur sampling means returning the classical spin label (j) with probability (p_j), and unitary Schur sampling means returning (j) together with the corresponding irrep state (\sigma_j) [2508.05046].

The protocol repeatedly performs a two-qubit SWAP test on a uniformly random pair among the active qubits. Using
[
\Pi_{\mathrm{sym}}=\frac{I+\mathrm{SWAP}}{2}=I-\xi,
\qquad
\Pi_{\mathrm{asym}}=\frac{I-\mathrm{SWAP}}{2}=\xi,
]
with (\xi=|\xi\rangle\langle\xi|) the singlet projector, each detected singlet is removed and recorded. If (k) singlets have been found, the classical register stores
[
|n/2-k\rangle,
]
which is interpreted as the current estimate (j'=n/2-k). The representation-theoretic reason this realizes weak Schur sampling is that singlets are spin-(0), so peeling them off does not change the total spin carried by the remaining qubits. For a state already in a fixed spin sector (j), the number of removable singlets is exactly (n/2-j) [2508.05046].

The paper proves asymptotic convergence to the Schur-transform channel on permutation-invariant inputs:
[
\lim_{T\to\infty}
\mathcal E_{\mathrm{SWAP}}^T

\left(\sigma\otimes |n/2\rangle\langle n/2|\right)

\mathcal E_{\mathrm{Schur}}(\sigma).
]
For a fixed (j)-sector input (\tau_j), the trace-distance error equals the probability that the register has not yet reached the correct (j):
[
D\Big(
\mathcal E_{\mathrm{Schur}}(\tau_j),\,
\mathcal E_{\mathrm{SWAP}}^T(\tau_j\otimes |n/2\rangle\langle n/2|)

\Big)

\Pr(j'\neq j;T).
]
More generally,
[
D\Big(
\mathcal E_{\mathrm{Schur}}(\sigma),\,
\mathcal E_{\mathrm{SWAP}}^T(\sigma\otimes |n/2\rangle\langle n/2|)
\Big)
\le
n\,\exp!\left(-\frac{T}{2n}\right),
\qquad
T\ge n\ln n,
]
so error at most (\epsilon) is guaranteed once
[
T\ge 2n\ln(n\epsilon^{-1}).
]
The paper therefore states that the same random-SWAP protocol achieves weak Schur sampling and unitary Schur sampling with error (\epsilon) after only (2n\ln(n\epsilon^{-1})) SWAP tests, and presents this as a lossless method for extracting any information invariant under permutations of qubits [2508.05046].

This qubit construction is also tied to purification. For (\rho=(1-p)|\psi\rangle\langle\psi|+p\,\mathbb I/2), the paper shows that random SWAP tests achieve the same fidelity as the Schur transform, which is optimal, and that the finite-(T) fidelity gap is bounded by the same exponentially decaying term (n e^{-T/(2n)}). The implementation, however, is specialized to qubits, and the main rigorous convergence theorem is formulated for permutation-invariant inputs [2508.05046].

6. Adjacent simulation results, limitations, and open directions

Several nearby results are often conflated with weak Schur sampling but concern strictly different tasks. The 2018 paper on “Quantum Schur Sampling circuits” studies circuits of the form
[
U_{\mathrm{Sch}}^\dagger U_\pi U_{\mathrm{Sch}}
\quad\text{or more generally}\quad
U_{\mathrm{Sch}}^\dagger \Lambda U_\pi U_{\mathrm{Sch}},
]
with computational-basis measurement at the end. Its main theorem is a strong-simulation result: efficient classical additive approximation of transition amplitudes (\langle \phi\mid\psi\rangle), not a general weak simulator for the circuit output distribution. The same paper does show that the computational-basis measurement distribution of a single sequentially coupled Schur-basis state is classically sampleable in polynomial time, but it explicitly states that “finding a method for sampling their output distribution remains open” [1801.04795].

The other 2018 paper is even closer representation-theoretically but still does not analyze weak Schur sampling in the textbook sense. It studies (\mathrm{SU}(2)) Schur circuits whose measured output is the full sequentially coupled Schur-basis label ((\mathbf J,M)), where (\mathbf J) is a path in the angular-momentum branching diagram and (M) is the magnetic quantum number. Weak Schur sampling is recovered only by coarse-graining:
[

p_{\mathrm{weak}}(J)

\sum_{\mathbf J:\,\mathrm{end}(\mathbf J)=J}\sum_{M=-J}^J p(\mathbf J,M).
]
The paper proves that if the refined distribution (p(\mathbf J,M)) is (\epsilon)-approximately (t)-sparse, then it can be classically sampled in
[
\operatorname{poly}!\left(n,\frac1\epsilon,t\right)
]
time to (6\epsilon) error in total variational distance. It also gives a polynomial-time algorithm for finding heavy paths (\mathbf J). But the sparsity assumption is imposed on the full Schur-basis distribution, not on the coarse weak-Schur marginal over (J), and the paper states no theorem specifically about sampling only (J) [1809.05171].

The resulting boundary is precise. Weak Schur sampling is the coarse irrep-label measurement. Strong Schur sampling resolves the full Schur basis. Unitary Schur sampling retains the unitary-group register while tracing out multiplicity. Full Schur-basis output sampling for ((\mathbf J,M)) is a further refinement in the (\mathrm{SU}(2)) sequential-coupling realization. The recent literature establishes efficient implementations for several of these tasks under different structural assumptions, but it does not collapse them into a single equivalence class. A plausible implication is that the complexity of coarse-grained weak Schur sampling can differ materially from the complexity of refined Schur-basis sampling, because sparsity or tractability of the latter need not characterize the former [2410.15793] [1809.05171].

Open directions stated across these works remain aligned with that distinction. One line concerns extending measured or streaming constructions beyond the currently analyzed settings, including broader (\mathrm{SU}(d)) or mixed-Schur regimes and qudit analogues of the random-SWAP protocol. Another concerns the classical complexity of weak Schur sampling itself, independently of stronger labels such as multiplicity paths or full Schur-basis data. A further line, emphasized in the Schur-circuit simulation work, is to determine when refined Schur output distributions are actually approximately sparse in broad circuit families, since current weak-simulation theorems depend on that promise rather than on the weak-Schur marginal alone [2508.05046] [1809.05171].