The Quantum Paldus Transform: Efficient Circuits with Applications

• URL: http://arxiv.org/abs/2506.09151v1
• Date: Tue, 10 Jun 2025 18:05:31 GMT
• Title: The Quantum Paldus Transform: Efficient Circuits with Applications
• Authors: Jędrzej Burkat, Nathan Fitzpatrick,
• Abstract summary: We present the Quantum Paldus Transform: an efficient quantum algorithm for block-diagonalising fermionic, spin-free Hamiltonians in the second quantisation.<n>Our work can be seen as a generalisation of the quantum Schur transform for the second quantisation, made tractable by the Pauli exclusion principle.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 for the Paldus transform with $\mathcal{O}(d^3)$ Toffoli complexity, paving the way for significant advancements in quantum simulation on quantum computers enabled by the UGA paradigm.

Related papers

• Efficient Qudit Circuit for Quench Dynamics of $2+1$D Quantum Link Electrodynamics [0.565395466029518]
  We propose a resource-efficient method for simulating $2+1$D spin-$S$ $mathrmU(1)$ quantum link lattice gauge theories with dynamical matter.<n>By integrating out the matter fields through Gauss's law, we reformulate the quantum link model in a purely spin picture compatible with qudit encoding.<n>Our framework significantly reduces the number of quantum resources and gate count.
  arXiv  Detail & Related papers  (2025-07-16T19:16:49Z)
• Universal framework with exponential speedup for the quantum simulation of quantum field theories including QCD [0.0]
  We present a quantum simulation framework universally applicable to a wide class of quantum systems.<n>Specifically, we generalize an efficient quantum simulation protocol developed for bosonic theories.<n>Our protocols do not assume oracles, but rather present explicit constructions with rigorous resource estimations.
  arXiv  Detail & Related papers  (2025-06-23T18:00:00Z)
• Fault-tolerant fermionic quantum computing [39.58317527488534]
  We introduce fermionic fault-tolerant quantum computing, a framework which removes this overhead altogether.<n>We show how our framework can be implemented in neutral atoms, overcoming the apparent inability of neutral atoms to implement non-number-conserving gates.
  arXiv  Detail & Related papers  (2024-11-13T19:00:02Z)
• Hybrid Oscillator-Qubit Quantum Processors: Simulating Fermions, Bosons, and Gauge Fields [31.51988323782987]
  We develop a hybrid oscillator-qubit processor framework for quantum simulation of strongly correlated fermions and bosons. This framework gives exact decompositions of particle interactions as well as approximate methods based on the Baker-Campbell Hausdorff formulas. While our work focusses on an implementation in superconducting hardware, our framework can also be used in trapped ion, and neutral atom hardware.
  arXiv  Detail & Related papers  (2024-09-05T17:58:20Z)
• Simulating electronic structure on bosonic quantum computers [34.84696943963362]
  We propose an approach to map the electronic Hamiltonian into a qumode bosonic problem that can be solved on bosonic quantum devices.<n>This work establishes a new pathway for simulating many-fermion systems, highlighting the potential of hybrid qubit-qumode quantum devices.
  arXiv  Detail & Related papers  (2024-04-16T02:04:11Z)
• Quantum convolutional channels and multiparameter families of 2-unitary matrices [0.0]
  We present a novel approach to construct quantum channels with large entangling capacities inspired by convolution. In particular, we identify conditions necessary for the convolutional channels constructed using our method to possess maximal entangling power. We establish new, continuous classes of bipartite 2-unitary matrices of dimension $d2$ for $d = 7$ and $d = 9$, with $2$ and $4$ free nonlocal parameters.
  arXiv  Detail & Related papers  (2023-12-29T18:14:56Z)
• Variational-quantum-eigensolver-inspired optimization for spin-chain work extraction [39.58317527488534]
  Energy extraction from quantum sources is a key task to develop new quantum devices such as quantum batteries. One of the main issues to fully extract energy from the quantum source is the assumption that any unitary operation can be done on the system. We propose an approach to optimize the extractable energy inspired by the variational quantum eigensolver (VQE) algorithm.
  arXiv  Detail & Related papers  (2023-10-11T15:59:54Z)
• Gelfand-Tsetlin basis for partially transposed permutations, with applications to quantum information [0.9208007322096533]
  We study representation theory of the partially transposed permutation matrix algebra. We show how to simplify semidefinite optimization problems over unitary-equivariant quantum channels. We derive an efficient quantum circuit for implementing the optimal port-based quantum teleportation protocol.
  arXiv  Detail & Related papers  (2023-10-03T17:55:10Z)
• Quantum representation of finite groups [0.0]
  The concept of quantum representation of finite groups (QRFG) has been a fundamental aspect of quantum computing for quite some time. We provide a formal definition of this concept using both group theory and differential geometry. Our work proves the existence of a quantum representation for any finite group and outlines two methods for translating each generator of the group into a quantum circuit.
  arXiv  Detail & Related papers  (2022-09-29T18:01:03Z)
• Efficient Bipartite Entanglement Detection Scheme with a Quantum Adversarial Solver [89.80359585967642]
  Proposal reformulates the bipartite entanglement detection as a two-player zero-sum game completed by parameterized quantum circuits. We experimentally implement our protocol on a linear optical network and exhibit its effectiveness to accomplish the bipartite entanglement detection for 5-qubit quantum pure states and 2-qubit quantum mixed states.
  arXiv  Detail & Related papers  (2022-03-15T09:46:45Z)
• Quantum algorithms for grid-based variational time evolution [36.136619420474766]
  We propose a variational quantum algorithm for performing quantum dynamics in first quantization. Our simulations exhibit the previously observed numerical instabilities of variational time propagation approaches.
  arXiv  Detail & Related papers  (2022-03-04T19:00:45Z)
• Halving the cost of quantum multiplexed rotations [0.0]
  We improve the number of $T$ gates needed for a $b$-bit approximation of a multiplexed quantum gate with $c$ controls. Our results roughly halve the cost of state-of-art electronic structure simulations based on qubitization of double-factorized or tensor-hypercontracted representations.
  arXiv  Detail & Related papers  (2021-10-26T06:49:44Z)
• Realization of arbitrary doubly-controlled quantum phase gates [62.997667081978825]
  We introduce a high-fidelity gate set inspired by a proposal for near-term quantum advantage in optimization problems. By orchestrating coherent, multi-level control over three transmon qutrits, we synthesize a family of deterministic, continuous-angle quantum phase gates acting in the natural three-qubit computational basis.
  arXiv  Detail & Related papers  (2021-08-03T17:49:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.