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The Quantum Paldus Transform: Efficient Circuits with Applications

Jędrzej Burkat, Nathan Fitzpatrick

TL;DR

The Paldus transform lends tools from the UGA readily applicable to quantum computational chemistry, leading to maximally sparse representations of spin-free Hamiltonians, efficient preparation of Configuration State Functions, and a direct interpretation of quantum chemistry reduced density matrix elements in terms of $SU(2)$ angular momentum coupling.

Abstract

We present the Quantum Paldus Transform: an efficient quantum algorithm for block-diagonalising fermionic, spin-free Hamiltonians in the second quantisation. Our algorithm implements an isometry between the occupation number basis of a fermionic Fock space of $2d$ modes, and the Gelfand-Tsetlin (GT) states spanning irreducible representations of the group $U(d) \times SU(2)$. The latter forms a basis indexed by well-defined values of total particle number $N$, global spin $S$, spin projection $M$, and $U(d)$ GT patterns. This realises the antisymmetric unitary-unitary duality discovered by Howe and developed into the Unitary Group Approach (UGA) for computational chemistry by Paldus and Shavitt in the 1970s. The Paldus transform lends tools from the UGA readily applicable to quantum computational chemistry, leading to maximally sparse representations of spin-free Hamiltonians, efficient preparation of Configuration State Functions, and a direct interpretation of quantum chemistry reduced density matrix elements in terms of $SU(2)$ angular momentum coupling. The transform also enables the encoding of quantum information into novel Decoherence-Free Subsystems for use in communication and error mitigation. Our work can be seen as a generalisation of the quantum Schur transform for the second quantisation, made tractable by the Pauli exclusion principle. Alongside self-contained derivations of the underlying dualities we provide fault-tolerant circuit compilation methods with full gate counts for the Paldus transform, resulting in $\mathcal{O}(d^3)$ Toffoli complexity, where a transform on $50$ spatial orbitals would require a modest $5500$ Toffoli gates. This paves the way for significant advancements in quantum simulation on quantum computers enabled by the UGA paradigm.

The Quantum Paldus Transform: Efficient Circuits with Applications

TL;DR

The Paldus transform lends tools from the UGA readily applicable to quantum computational chemistry, leading to maximally sparse representations of spin-free Hamiltonians, efficient preparation of Configuration State Functions, and a direct interpretation of quantum chemistry reduced density matrix elements in terms of angular momentum coupling.

Abstract

We present the Quantum Paldus Transform: an efficient quantum algorithm for block-diagonalising fermionic, spin-free Hamiltonians in the second quantisation. Our algorithm implements an isometry between the occupation number basis of a fermionic Fock space of modes, and the Gelfand-Tsetlin (GT) states spanning irreducible representations of the group . The latter forms a basis indexed by well-defined values of total particle number , global spin , spin projection , and GT patterns. This realises the antisymmetric unitary-unitary duality discovered by Howe and developed into the Unitary Group Approach (UGA) for computational chemistry by Paldus and Shavitt in the 1970s. The Paldus transform lends tools from the UGA readily applicable to quantum computational chemistry, leading to maximally sparse representations of spin-free Hamiltonians, efficient preparation of Configuration State Functions, and a direct interpretation of quantum chemistry reduced density matrix elements in terms of angular momentum coupling. The transform also enables the encoding of quantum information into novel Decoherence-Free Subsystems for use in communication and error mitigation. Our work can be seen as a generalisation of the quantum Schur transform for the second quantisation, made tractable by the Pauli exclusion principle. Alongside self-contained derivations of the underlying dualities we provide fault-tolerant circuit compilation methods with full gate counts for the Paldus transform, resulting in Toffoli complexity, where a transform on spatial orbitals would require a modest Toffoli gates. This paves the way for significant advancements in quantum simulation on quantum computers enabled by the UGA paradigm.

Paper Structure

This paper contains 61 sections, 25 theorems, 233 equations, 57 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Summary of Results and Outline
  3. Background
  4. Second Quantisation and the Exterior Product Space
  5. Schur Polynomials and Semi-Standard Young Tableaux
  6. Antisymmetric Unitary-Unitary Duality
  7. Paldus Duality
  8. Subgroup Branching and Shavitt Graphs
  9. SU(2) Spin Coupling and the Gelfand-Tsetlin Basis
  10. Explicit Construction of Gelfand-Tsetlin States
  11. Spin Coupling as Rotations
  12. The Quantum Paldus Transform
  13. Clebsch-Gordan Transforms for the Paldus Transform
  14. Incrementing the Spin Projection Register (M)
  15. Controlled Givens Rotations
  16. ...and 46 more sections

Key Result

Theorem 1

The irreducible decomposition of the exterior power of the tensor product space $\mathbb{C}^m \otimes \mathbb{C}^n$ on which the elements of $U(n) \otimes U(m)$ act is given by: where $U(mn) \downarrow U(m) \times U(n)$ refers to the restriction of $U(mn)$ to its subgroup $U(m) \times U(n)$, and $W_{\lambda_k}$ and $W_{\tilde{\lambda}_k}$ are the irreducible representations of $U(m)$ and $U(n)$ i

Figures (57)

  • Figure 1: $U(d) \times SU(2)$ irrep $(N=6,S=1)$, which is expressed by the partition $\lambda = (4,2)$ and its conjugate $\tilde{\lambda} = (2,2,1,1)$. $N$ is calculated from the number of boxes in both partitions and the $S$ is calculated from the difference between the number of boxes in the first and second row of $\lambda$, given by $S = \frac{1}{2}(\lambda_1 - \lambda_2) = \frac{1}{2}(4-2) = 1$.
  • Figure 2: Left: Young Diagram representing the $d$-part partition $\lambda = (2, \hdots ,2,1,\hdots,1,0,\hdots,0)$, which is abbreviated to $(2^a, 1^b, 0^c)$. The restriction to two columns imposed by Theorem \ref{['thm:paldus_duality']} allows for a compact representation of $\lambda$ through the tuple $(a, b, c)$. Right: Irreducible representations of in the antisymmetric $U(3) \times U(2)$ representation. Each Young diagram denotes an $(N, S)$ irrep of $U(3)$. $U(2)$ irrep labels are obtained by conjugating the Young diagrams.
  • Figure 3: A superposition of all Shavitt graphs for $d=3$. The nodes denote $(a_i,b_i,c_i)$ triples defining the irreps at each link of the subgroup chain. The steps $\mathbf{d}_i$ are shown by the four coloured arcs, with a full step vector $\mathbf{d}$ consisting of a walk from a $d=0$ (tail) to the $d=3$ (head) nodes. The Gelfand-Tsetlin basis states spanning the irrep $\lambda$ are defined by unique paths from head to tail of the Shavitt graph, which is equivalent to the unique subduction chain $U(d) \times U(2) \supset \cdots \supset U(1) \times U(2)$.
  • Figure 4: Shavitt graph for the $U(3) \times U(2)$ irrep $\lambda = (1,1,1)$ via the subgroup chain $U(3)\times U(2) \supset U(2) \times U(2) \supset U(1) \times U(2)$. Each walk denotes a Gelfand-Tsetlin state, with the associated semi-standard Young tableau also shown. The collection of all walks leads to a Gelfand-Tsetlin basis of the $U(3)$ irrep.
  • Figure 5: Top left: Coupling by steps $\mathbf{d}_i \in \{0,1,2,3\}$ out of a node $N, S$ of a Shavitt graph. The $(\Delta S, \Delta N)$ for each $\mathbf{d}_i$ are represented inside the nodes. Top right: The corresponding coupling of $M$ values which appear in the spin coupling graphs of Figure \ref{['fgr:mz_gt_coupling']}. Bottom: The overall effect of the coupling can be seen as a rotation from tensor product states $\ket{N, S, M'; \mathbf{d}} \otimes \ket{m_i}$ (bottom nodes) into Gelfand-Tsetlin states $\ket{N', S', M; \mathbf{d}'}$ (top nodes).
  • ...and 52 more figures

Theorems & Definitions (93)

  • Definition 1: SSYT Schur Polynomial
  • Example
  • Definition 2: Standard Irrep of $U(d)$
  • Theorem 1: Antisymmetric Unitary-Unitary Duality
  • proof
  • Remark
  • Theorem 2: Paldus Duality
  • proof
  • Remark
  • Definition 3: Quantum Paldus Transform
  • ...and 83 more