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arXiv:2312.14909

Permutation-invariant quantum circuits

Maximilian Balthasar Mansky, Santiago Londoño Castillo, Victor Ramos Puigvert, Claudia Linnhoff–Popien

Published Dec 22, 2023• 1 citation
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Abstract

The implementation of physical symmetries into problem descriptions allows for the reduction of parameters and computational complexity. We show the integration of the permutation symmetry as the most restrictive discrete symmetry into quantum circuits. The permutation symmetry is the supergroup of all other discrete groups. We identify the permutation with a $\operatorname{SWAP}$ operation on the qubits. Based on the extension of the symmetry into the corresponding Lie algebra, quantum circuit element construction is shown via exponentiation. This allows for ready integration of the permutation group symmetry into quantum circuit ansatzes. The scaling of the number of parameters is found to be $\mathcal{O}(n^3)$, significantly lower than the general case and an indication that symmetry restricts the applicability of quantum computing. We also show how to adapt existing circuits to be invariant under a permutation symmetry by modification.

DOI
10.48550/arxiv.2312.14909

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The implementation of physical symmetries into problem descriptions allows for the reduction of parameters and computational complexity. We show the integration of the permutation symmetry as the most restrictive discrete symmetry into quantum circuits. The permutation symmetry is the supergroup of all other discrete groups. We identify the permutation with a SWAP⁡\operatorname{SWAP}SWAP operation on the qubits. Based on the extension of the symmetry into the corresponding Lie algebra, quantum circuit element construction is shown via exponentiation. This allows for ready integration of the permutation group symmetry into quantum circuit ansatzes. The scaling of the number of parameters is found to be O(n3)\mathcal{O}(n^3)O(n3), significantly lower than the general case and an indication that symmetry restricts the applicability of quantum computing. We also show how to adapt existing circuits to be invariant under a permutation symmetry by modification.
We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group SU(2n)SU(2^n)SU(2n), in particular composition, algebraic and topological closedness and connectedness. It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space. The well-behavedness is independent of the symmetry requirement imposed on the subgroup. We provide an example of a permutation invariance across all qubits.
The structures are physical systems and/or their mathematical descriptions consisting of elements such as particles, fields, observables, operators etc and of relations between these elements: the equations of motion and the rules of quantum mechanics. Examples of symmetries are reflections, permutations of particles, time reversal. The physical system has a symmetry group means, then, that there is a group of transformations leaving the equations of motion and the rules of quantum mechanics invariant. The discrete symmetries lead to conservation laws frequently expressible in terms of quantum numbers, but sometimes requiring a more elaborate description. The observables corresponding to the super-selection quantum numbers can be measured without disturbing any other measurement; their values are always sharp. The chapter shows that the invariant subspace may be labelled by continuous quantum numbers (for instance momentum) when the direct sum becomes a direct integral.
Symmetry is fundamental in the description and simulation of quantum systems. Leveraging symmetries in classical simulations of many-body quantum systems can result in significant overhead due to the exponentially growing size of some symmetry groups as the number of particles increases. Quantum computers hold the promise of achieving exponential speedup in simulating quantum many-body systems; however, a general method for utilizing symmetries in quantum simulations has not yet been established. In this work, we present a unified framework for incorporating symmetry group transforms on quantum computers to simulate many-body systems. The core of our approach lies in the development of efficient quantum circuits for symmetry-adapted projection onto irreducible representations of a group or pairs of commuting groups. We provide resource estimations for common groups, including the cyclic and permutation groups. Our algorithms demonstrate the capability to prepare coherent superpositions of symmetry-adapted states and to perform quantum evolution across a wide range of models in condensed-matter physics and ab initio electronic structure in quantum chemistry. Specifically, we execute a symmetry-adapted quantum subroutine for small molecules in first-quantization on noisy hardware and demonstrate the emulation of symmetry-adapted quantum phase estimation for preparing coherent superpositions of quantum states in various irreducible representations of a symmetry group. In addition, we present a discussion of open problems regarding treating symmetries in digital quantum simulations of many-body systems, paving the way for future systematic investigations into leveraging symmetries quantumly for practical quantum advantage. The broad applicability and rigorous resource estimation for symmetry transformations make our framework appealing for achieving provable quantum advantage on fault-tolerant quantum computers, especially for symmetry-related properties.
This project is a comprehensive investigation into the application of the exchange interaction, particularly with the realization of the SWAP^1/n quantum operator, in quantum information processing. We study the generation, characterization and application of entanglement in such systems. Given the non-commutativity of neighbouring SWAP^1/n gates, the mathematical study of combinations of these gates is an interesting avenue of research that we have explored, though due to the exponential scaling of the complexity of the problem with the number of qubits in the system, numerical techniques, though good for few-qubit systems, are found to be inefficient for this research problem when we look at systems with higher number of qubits. Since the group of SWAP^1/n operators is found to be isomorphic to the symmetric group Sn, we employ group-theoretic methods to find the relevant invariant subspaces and associated vector-states. Some interesting patterns of states are found including onedimensional invariant subspaces spanned by W-states and the Hamming-weight preserving symmetry of the vectors spanning the various invariant subspaces. We also devise new ways of characterizing entanglement and approach the separability problem by looking at permutation symmetries of subsystems of quantum states. This idea is found to form a bridge with the entanglement characterization tool of Peres-Horodecki’s Partial Positive Transpose (PPT), for mixed quantum states. We also look at quantum information taskoriented ‘distance’ measures of entanglement, besides devising a new entanglement witness in the ‘engle’. In terms of applications, we define five different formalisms for quantum computing: the circuit-based model, the encoded qubit model, the cluster-state model, functional quantum computation and the qudit-based model. Later in the thesis, we explore the idea of quantum computing based on decoherence-free subspaces. We also investigate ways of applying the SWAP^1/n in entanglement swapping for quantum repeaters, quantum communication protocols and quantum memory.
We introduce a practical construction of group-equivariant and permutation-invariant functions of NNN variables given a finite-dimensional space stable with respect to the group action. The construction applies to any connected linear Lie group and relies on leveraging the Lie algebra to build a matrix MMM whose kernel is in one-to-one correspondence with the subspace with desired equivariance and invariance properties, removing the need for prior knowledge of Clebsch--Gordan coefficients. A similar construction is proposed for group-equivariant functions alone, without imposing permutation-invariance. For the groups SO(3)SO(3)SO(3) and SU(2)SU(2)SU(2), we further exploit the structure of the Lie algebra to demonstrate the sparsity pattern and rank of the matrix MMM, which yields the exact dimension of the group-equivariant and permutation-invariant space, as well as the dimension of the group-equivariant space alone. We demonstrate analytically and verify numerically that the proposed method scales linearly with respect to the dimensionality of the basis, offering a high computational gain compared to existing methods in the literature which typically scale exponentially. We finally perform a dimensionality comparison, showing that for large values of~NNN, the dimension of group-equivariant and permutation-invariant spaces is of comparable order as the dimension of permutation-invariant spaces, while pre-asymptotically, the first dimensionality is orders of magnitude lower than the second. Hence a substantial computational gain can be achieved by explicitly enforcing group-equivariance on top of permutation-invariance when approximating such functions.
Abstract Symmetries are omnipresent in physics and have been used to reduce the number of degrees of freedom of systems. In this work, we investigate the properties of M SU ( n ) , M -invariant subspaces of the special unitary Lie group SU(n). This group is relevant to quantum computing and quantum systems in general. We demonstrate that for certain choices of M , the subset M SU ( n ) inherits many topological and group properties from SU(n). We then present a combinatorial method for computing the dimension of such subspaces when M is a representation of a permutation group acting on a tensor product of spaces G SU ( n ) , or a Hamiltonian H ( n ) SU ( n ) . The Kronecker product of su ( 2 ) matrices is employed to construct the Lie algebras associated with different permutation-invariant groups GSU(n). Numerical results on the number of dimensions support the developed theory.