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Reducing qubit requirements for quantum simulation using molecular point group symmetries
Kanav Setia, Richard Chen, Julia E. Rice, Antonio Mezzacapo, Marco Pistoia, James Whitfield
TL;DR
Because qubits are limited on near-term quantum computers, the paper develops a way to use molecular spatial symmetries in simulation. It maps point-group symmetries from second quantization to qubit operators and uses them to taper qubits, finding additional reductions for some molecules compared with generic Z2 symmetries. The authors conclude that the formalism can spare more qubits in the mapping to a quantum computer.
Problem
Qubits remain limited on near-term devices, and the physical significance of many symmetries found by generic Z2 tapering was unresolved.
Source: §I · p. 1, §II · p. 3Method
The paper represents molecular point-group symmetries in second quantization, maps them to qubit operators, and uses suitable Pauli symmetries to taper qubits.
Source: §I · p. 1, §V · pp. 4–5Results
For BeH2, point-group symmetries taper five qubits, compared with four using the method of [13].
Source: BeH2 (Symmetry group:D∞h) · p. 5Takeaways & Limitations
The formalism can identify additional molecular symmetries and spare more qubits than one result reported in [13].
Source: §VIII · p. 6
Abstract
from arXiv · showhide
Simulating molecules is believed to be one of the early-stage applications for quantum computers. Current state-of-the-art quantum computers are limited in size and coherence, therefore optimizing resources to execute quantum algorithms is crucial. In this work, we develop the second quantization representation of the spatial-symmetries which are then transformed to their qubit operator representation. These qubit operator representations are used to reduce the number of qubits required for simulating molecules. We present our results for various molecules and elucidate a formal connection of this work with a previous technique that analyzed generic $Z_2$ Pauli symmetries.
I. INTRODUCTION
Because qubits remain scarce on near-term devices, this work uses molecular symmetries to reduce simulation qubit requirements. It connects spatial point-group symmetries to qubit tapering and gives physical meaning to some symmetries found by earlier generic Z2 methods.
I. INTRODUCTION
- The paper places its approach alongside prior quantum-chemistry methods and notes that some fermionic encodings use more qubits than fermionic modes.The introduction also cites quantum simulation of BeH2 and development of trapped-ion devices.Source: §I · p. 1
- Qubits remain precious on NISQ devices, motivating methods that reduce the number needed to simulate a molecular system.The paper frames resource optimization as important while fault-tolerant quantum computers remain some time away.Source: §I · p. 1
- Earlier Z2 tapering found Hamiltonian Pauli symmetries but left the physical meaning of many such operators unclear.The work asks whether these symmetries connect to molecular point-group symmetries.Source: §I · p. 1
- The paper develops second-quantized representations of molecular point-group symmetries, maps them to qubit operators, and uses suitable commuting Pauli strings to taper qubits.It also presents a way to identify the correct eigensector, reducing the experiments needed compared with the previous approach.Source: §I · p. 1
- The paper proceeds from a review of Z2 tapering through point-group formalism and diagonalization to results for molecules in different point groups.Its stated sequence covers Sections II–VI.Source: §I · p. 2
II. QUBIT TAPERING BASED ON Z2 SYMMETRIES
Qubit tapering exploits commuting Pauli symmetries to remove qubits from a molecular simulation. The section describes how to find those symmetries and transform the Hamiltonian, and motivates the search for their physical connection to molecular point groups.
II. QUBIT TAPERING BASED ON Z2 SYMMETRIES
- The Hamiltonian is mapped from fermionic operators to Pauli strings, while its one- and two-body integrals specify the molecular Hamiltonian.The text identifies Jordan–Wigner and parity as example mappings and notes Bravyi–Kitaev among common encodings.Source: §II · p. 2
- The kernel of the check matrix E yields the symmetry group, whose generators can be obtained by Gram–Schmidt orthogonalization over Z2.The Pauli strings are represented as binary vectors; commutation is tested by axbz + azbx = 0 mod 2.Source: §II · pp. 2–3
- Unitary transformations and qubit permutations convert the symmetry generators into single-qubit Pauli operators and transform the Hamiltonian accordingly.The construction seeks an X operator that anticommutes with one generator and commutes with the others.Source: §II · pp. 2–3
- A qubit can be removed when every Hamiltonian term acts on it trivially or with at most one Pauli operator, whose eigenvalue replaces that operator.After transformation, the last k qubits can be tapered by substituting the corresponding σx eigenvalues, ±1.Source: §II · pp. 2–3
- For LiH, H2O, and BeH2, earlier Z2 tapering removed 4, 4, and 3 qubits, respectively, across the explored encodings.Those results used standard geometries and the STO-3G basis set.Source: §II · p. 3
- The unresolved question motivating this work is the physical significance of tapering symmetries and their possible connection to molecular point groups.The earlier procedure could identify symmetries whose physical interpretation was not obvious.Source: §II · p. 3
III. POINT GROUP SYMMETRIES
Molecular point groups comprise spatial operations that leave a molecule unchanged. The paper uses representations of these symmetries, including those built from orbital functions, as a route to qubit tapering.
III. POINT GROUP SYMMETRIES
- The four required nontrivial point-group operations are proper rotation, plane reflection, improper rotation, and inversion.Inversion is equivalent to S2.Source: §III · p. 3
- A molecule’s point group is determined by the symmetry elements that leave it unchanged; for example, H2O is C2v and NH3 is C3v.Character-table rows denote irreducible representations and columns group symmetry elements into classes.Source: §III · p. 3
- Representations built from atomic- or molecular-orbital functions, as in symmetry-adapted linear combinations, can also be used to taper qubits.The paper contrasts this with vector representations of points in space and notes that orbital-function representations traditionally build symmetry-adapted molecular orbitals.Source: §III · p. 3
IV. POINT GROUP SYMMETRIES IN SECOND QUANTIZATION
The paper represents molecular spatial symmetries in second quantization and maps them to qubit operators. It checks symmetry against the Hamiltonian and notes that signed-permutation matrices make the second-quantized construction tractable.
IV. POINT GROUP SYMMETRIES IN SECOND QUANTIZATION
- A point-group operation is represented by a unitary matrix on a truncated set of single-particle wave functions, with the matrices following the group multiplication table.The unitary condition preserves the canonical commutation relations after transforming the second-quantized operators.Source: §IV · p. 4
- Symmetry is verified by transforming the one- and two-body integral tensors and checking that the Hamiltonian remains unchanged.This checks commutation with the Hamiltonian without constructing its 2M × 2M representation.Source: §IV · p. 4
- Restricting the unitary to signed-permutation matrices makes its second-quantized representation straightforward to construct.Any permutation can be decomposed into transpositions, and the paper gives a second-quantized factor for the fermionic −1 phase.Source: §IV · p. 4
- Second-quantized representations of point-group operations can be mapped through a fermion-to-qubit transformation, generally producing sums of Pauli strings.The tapering procedure applies when a symmetry has a single-Pauli-string representation; the paper describes conditions and a procedure for reaching that form.Source: §IV · p. 4
V. REPRESENTING POINT GROUP SYMMETRIES AS PAULI OPERATORS
The section turns permutation symmetries of a molecular Hamiltonian into qubit symmetries that can support qubit tapering. For mode swaps, the transformed symmetry becomes Z-type under Jordan–Wigner, allowing an n-1-qubit simulation.
V. REPRESENTING POINT GROUP SYMMETRIES AS PAULI OPERATORS
- A Hamiltonian-invariant mode swap yields a Z-type symmetry under Jordan–Wigner, so the transformed Hamiltonian can be simulated with n-1 qubits.The transformed Hamiltonian retains its single-particle and two-particle operators when the original contains only those operators.Source: §V · pp. 4–5
- The construction represents a permutation by a unitary matrix, diagonalizes its Hermitian generator, and defines a symmetry that commutes with the transformed Hamiltonian.The transformation uses a unitary V that diagonalizes G; the resulting S commutes with H′.Source: §V · pp. 4–5
- Diagonalizing R directly can yield S without constructing G; its qubit operator applies σz where S has diagonal entry −1.Source: §V · p. 5
- Multiple symmetries can be simultaneously diagonalized only when they commute; otherwise, tapering uses a maximal commuting subset.Source: §V · p. 5
- The general qubit-operator representation of spatial symmetries need not be a single Pauli string, and the method identifies extra molecular symmetries beyond [13].Source: §V · p. 5
VI. RESULTS
Spatial symmetries yield additional qubit tapering beyond the method in [13] for BeH2, while ammonia’s noncommuting symmetry classes restrict tapering to one reflection. The section also illustrates how symmetries are constructed and how the correct eigensector is selected.
VI. RESULTS
- For H2 at 0.7414 Å in STO-3G, C2 and σ(yz) swap the two hydrogen 1s orbitals, yielding a permutation matrix whose G matrix is diagonalized for tapering.The Hamiltonian remains unchanged under the corresponding permutation of fermionic modes.Source: H2 (Symmetry group: D∞h) · p. 5
- Five qubits are tapered for linear BeH2 with STO-3G, versus four using [13].The geometry uses 1.291 Å bonds between beryllium and hydrogen.Source: BeH2 (Symmetry group:D∞h) · p. 5
- Table I tests molecules at equilibrium geometry with STO-3G and includes two tapered qubits from spin-up and spin-down conservation; all listed symmetries form an abelian group.The one- and two-body terms are obtained in the atomic-orbital basis using PySCF.Source: §VI · p. 5, Table I · p. 6
- For BeH2, σ(yz) yields a sum of Pauli strings and enables spatial-symmetry tapering, although the direct method in [13] does not find it.The σ(xy) and σ(xz) operators match symmetries found using [13].Source: BeH2 (Symmetry group:D∞h) · p. 6
- For NH3 in C3v, noncommuting symmetry classes permit tapering by only one symmetry, and the R-squared identity condition selects a reflection absent from [13].The two rotation operators form one class and the three reflections another.Source: NH3, (Symmetry group: C3v) · p. 6
VII. PICKING THE RIGHT EIGENVALUES FOR THE SYMMETRIES
- The correct symmetry eigensector follows from occupation parity: occupied and unoccupied Jordan–Wigner modes have σz eigenvalues −1 and +1, respectively.For BeH2, six electrons give three electrons in each spin sector, selecting the −1 eigenvalue for both spin-conservation symmetries.Source: §VII · p. 6
VI. RESULTS
- Molecular orbitals already symmetrized by irreducible representations give Z2 symmetries; constructing the R matrices manually is currently required for this formalism.The subroutine in [13] is more automated, while R matrices provide physical intuition and a way to verify expected symmetries.Source: §VII · p. 6
VIII. CONCLUSIONS
The conclusion presents a formalism for using molecular spatial symmetries to reduce simulation qubits. Compared with [13], it discovers additional symmetries and spares more qubits, with relevance to NISQ resource optimization.
VIII. CONCLUSIONS
- The formalism exploits spatial symmetries in molecular systems to reduce the number of qubits required for quantum simulations.Source: §VIII · p. 6
- Compared with [13], the work discovers additional symmetries and spares more qubits in mapping to a quantum computer.Source: §VIII · p. 6
- The results are relevant to optimizing resources in NISQ computing [1].Source: §VIII · p. 6
IX. ACKNOWLEGEMENTS
The acknowledgements name NSF support for KS and JDW, and additional support for JDW.
IX. ACKNOWLEGEMENTS
- KS and JDW are funded by NSF awards DMR-1747426 and 1820747.Source: §IX · p. 7
- JDW also acknowledges U.S. Department of Energy support under Award A053685.Source: §IX · p. 7
- JDW additionally acknowledges support from a Department.Source: §IX · p. 7