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arXiv:1205.3928

A quantum algorithm for the quantum Schur-Weyl transform

Sonya J. Berg

Published May 17, 2012• 2 citations
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Abstract

We construct an efficient quantum algorithm to compute the quantum Schur-Weyl transform for any value of the quantum parameter $q \in [0,\infty]$. Our algorithm is a $q$-deformation of the Bacon-Chuang-Harrow algorithm, in the sense that it has the same structure and is identically equal when $q=1$. When $q=0$, our algorithm is the unitary realization of the Robinson-Schensted-Knuth (or RSK) algorithm, while when $q=\infty$ it is the dual RSK algorithm together with phase signs. Thus, we interpret a well-motivated quantum algorithm as a generalization of a well-known classical algorithm.

DOI
10.48550/arxiv.1205.3928

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We construct an efficient quantum algorithm to compute the quantum Schur-Weyl transform for any value of the quantum parameter q∈[0,∞]q \in [0,\infty]q∈[0,∞]. Our algorithm is a qqq-deformation of the Bacon-Chuang-Harrow algorithm, in the sense that it has the same structure and is identically equal when q=1q=1q=1. When q=0q=0q=0, our algorithm is the unitary realization of the Robinson-Schensted-Knuth (or RSK) algorithm, while when q=∞q=\inftyq=∞ it is the dual RSK algorithm together with phase signs. Thus, we interpret a well-motivated quantum algorithm as a generalization of a well-known classical algorithm.
The Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on an n fold tensor product V ⊗ n of a vector space V of dimension d . Bacon, Chuang and Harrow [5] gave a quantum algorithm for this transform that is polynomial in n , d and log ⁡ ϵ − 1 , where ϵ is the precision. In a footnote in Harrow's thesis [18], a brief description of how to make the algorithm of [5] polynomial in log ⁡ d is given using the unitary group representation theory (however, this has not been explained in detail anywhere). In this article, we present a quantum algorithm for the Schur transform that is polynomial in n , log ⁡ d and log ⁡ ϵ − 1 using a different approach. Specifically, we build this transform using the representation theory of the symmetric group and in this sense our technique can be considered a ''dual" algorithm to [5]. A novel feature of our algorithm is that we construct the quantum Fourier transform over the so called permutation modules , which could have other applications.
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.
The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we extend the framework for the QFT from finite groups to finite-dimensional semisimple algebras, and give efficient QFTs for the partition algebra Pn(d)P_n(d)Pn​(d), Brauer algebra Bn(d)B_n(d)Bn​(d), and walled Brauer algebra Br,s(d)B_{r,s}(d)Br,s​(d). These algebras play important roles in generalized Schur-Weyl duality, statistical physics and many-body systems, and have recently found several applications in quantum algorithms. Unlike the group case, the Fourier transform over a semisimple algebra can be non-unitary. Nevertheless, we show that when the parameter ddd is sufficiently large, the Fourier transform is well approximated by a unitary operator. Furthermore, we show that for each of the algebras AAA from above, such an approximate Fourier transform can be implemented efficiently: we give a quantum algorithm with gate complexity poly(n,log⁡d,log⁡(1/ε))\mathrm{poly}(n,\log d,\log(1/\varepsilon))poly(n,logd,log(1/ε)) for approximating the Fourier transform to error (d−1/2+ε)⋅poly(∣A∣)(d^{-1/2} + \varepsilon) \cdot \mathrm{poly}(|A|)(d−1/2+ε)⋅poly(∣A∣). Along the way, we establish several properties of the Fourier basis of semisimple algebras that may be of independent interest.
We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d-dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the Schur transform to an accuracy of e-1, then our circuit construction uses a number of gates which is polynomial in n, d and log(e-1). The key tool in our construction is a poly(d, log n, log(e-1)) algorithm for the ud Clebsch-Gordan transform. Our efficient circuit construction renders numerous protocols in quantum information theory computationally tractable and yields a new possible approach to quantum algorithms which is distinct from the standard paradigm of the quantum Fourier transform.
We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of (Cd)⊗n(\mathbb{C}^d)^{\otimes n}(Cd)⊗n and the Schur-Weyl basis of (Cd)⊗n(\mathbb{C}^d)^{\otimes n}(Cd)⊗n. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a major improvement on Louck's formula for calculating the transition amplitudes between two standard Weyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph for the case d=2d=2d=2, merely by looking at the entries in the two tableaux. The key theoretical component that underpins our results is the discovery of a branching rule for the Schur-Weyl states, which we call the Schur-Weyl branching rule. This branching rule allows us to perform the change of basis transformation described above in a straightforward manner for any nnn and ddd.