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arXiv:2309.11947

Weak Schur sampling with logarithmic quantum memory

Enrique Cervero-Martí­n, Laura Mančinska

Published Sep 21, 2023• 0 citations
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Abstract

The quantum Schur transform maps the computational basis of a system of $n$ 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 $n$ $d$-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 $n$ qubits to accuracy $ε$ requires only $O(\log_2n)$ qubits of memory and $O(n^3\log_2(\frac{n}ε))$ gates from the Clifford+T set. Further, we show that our weak Schur sampling algorithm on $n$ qudits decomposes into $O\big(dn^{2d}\log_2^p\big(\frac{n^{2d}}ε\big)\big)$ gates from an arbitrary fault-tolerant qudit universal set, for $p\approx 4$, and requires a memory of $O(\log_dn)$ qudits to implement.

DOI
10.48550/arxiv.2309.11947

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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.
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.
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We describe an efficient quantum algorithm for the quantum Schur transform. The Schur transform is an operation on a quantum computer that maps the standard computational basis to a basis composed of irreducible representations of the unitary and symmetric groups. We simplify and extend the algorithm of Bacon, Chuang, and Harrow, and provide a new practical construction as well as sharp theoretical and practical analyses. Our algorithm decomposes the Schur transform on n qubits into O(n^4\log(\frac{n}{\eps})) operators in the Clifford+T fault-tolerant gate set and uses exactly 2\lfloor\log_2(n)\rfloor-1 ancillary qubits. We extend our qubit algorithm to decompose the Schur transform on n qudits of dimension d into O(d^{1+p}n^{3d}\log^p(\frac{d n}{\eps})) primitive operators from any universal gate set, for p~3.97.
The quantum Schur transform is a fundamental building block that maps the computational basis to a coupled basis consisting of irreducible representations of the unitary and symmetric groups. Equivalently, it may be regarded as a change of basis from the computational basis to a simultaneous spin eigenbasis of Permutational Quantum Computing (PQC) [Quantum Inf. Comput., 10, 470-497 (2010)]. By adopting the latter perspective, we present a transparent algorithm for implementing the qubit quantum Schur transform which uses O(log⁡(n))O(\log(n))O(log(n)) ancillas and can be decomposed into a sequence of O(n3log⁡(n)log⁡(nε))O(n^3\log(n)\log(\frac{n}ε))O(n3log(n)log(εn​)) Clifford + T gates, where εεε is the accuracy of the algorithm in terms of the trace norm. We discuss the necessity for some applications of implementing this operation as a unitary rather than an isometry, as is often presented. By studying the associated Schur states, which consist of qubits coupled via Clebsch-Gordan coefficients, we introduce the notion of generally coupled quantum states. We present six conditions, which in different combinations ensure the efficient preparation of these states on a quantum computer or their classical simulability (in the sense of computational tractability). It is shown that Wigner 6-j symbols and SU(N) Clebsch-Gordan coefficients naturally fit our framework. Finally, we investigate unitary transformations which preserve the class of computationally tractable states.
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.
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.