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arXiv:quant-ph/0407082 · Physical Review Letters

Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms

Dave Bacon, Isaac L. Chuang, Aram W. Harrow

Published Oct 27, 2006• 162 citations
arXivHTMLPDF

Abstract

The Schur basis on n d-dimensional quantum systems is a generalization of the total angular momentum basis that is useful for exploiting symmetry under permutations or collective unitary rotations. We present efficient {size poly[n,d,log(1/epsilon)] for accuracy epsilon} quantum circuits for the Schur transform, which is the change of basis between the computational and the Schur bases. Our circuits provide explicit efficient methods for solving such diverse problems as estimating the spectrum of a density operator, quantum hypothesis testing, and communicating without a shared reference frame. We thus render tractable a large series of methods for extracting resources from quantum systems and for numerous quantum information protocols.

DOI
10.1103/physrevlett.97.170502

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The Schur basis on n d-dimensional quantum systems is a generalization of the total angular momentum basis that is useful for exploiting symmetry under permutations or collective unitary rotations. We present efficient {size poly[n,d,log(1/epsilon)] for accuracy epsilon} quantum circuits for the Schur transform, which is the change of basis between the computational and the Schur bases. Our circuits provide explicit efficient methods for solving such diverse problems as estimating the spectrum of a density operator, quantum hypothesis testing, and communicating without a shared reference frame. We thus render tractable a large series of methods for extracting resources from quantum systems and for numerous quantum information protocols.
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 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 epsilon, then our circuit construction uses a number of gates which is polynomial in n, d and log(1/epsilon). The important insights we use to perform this construction are the selection of the appropriate subgroup adapted basis and the Wigner-Eckart theorem. Our efficient circuit construction renders numerous protocols in quantum information theory computationally tractable and is an important new efficient quantum circuit family which goes significantly beyond the standard paradigm of the quantum Fourier transform.
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 Schur transform, which block-diagonalizes the tensor representation U⊗nU^{\otimes n}U⊗n of the unitary group Ud\mathbf{U}_dUd​ on nnn qudits, is an important primitive in quantum information and theoretical physics. We give a generalization of its quantum circuit implementation due to Bacon, Chuang, and Harrow (SODA 2007) to the case of mixed tensor U⊗n⊗Uˉ⊗mU^{\otimes n} \otimes \bar{U}^{\otimes m}U⊗n⊗Uˉ⊗m, where Uˉ\bar{U}Uˉ is the dual representation. This representation is the symmetry of unitary-equivariant channels, which find various applications in quantum majority vote, multiport-based teleportation, asymmetric state cloning, black-box unitary transformations, etc. The "mixed" Schur transform contains several natural extensions of the representation theory used in the Schur transform, in which the main ingredient is a duality between the mixed tensor representations and the walled Brauer algebra. Another element is an efficient implementation of a "dual" Clebsch-Gordan transform for Uˉ\bar{U}Uˉ. The overall circuit has complexity O~((n+m)d4)\widetilde{O} ((n+m)d^4)O((n+m)d4). Finally, we show how the mixed Schur transform enables efficient implementation of unitary-equivariant channels in various settings and discuss other potential applications, including an extension of permutational quantum computing that includes partial transposes.
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.
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.