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Pauli Partitioning with Respect to Gate Sets
Andrew Jena, Scott Genin, Michele Mosca
TL;DR
The paper studies how to partition Pauli operators into groups that can be measured simultaneously, motivated by reducing VQE measurement work. It formalizes the partitioning problem through graph coloring and compares gate-set choices. It reports a conjectured linear-in-length reduction in the expected number of parts with arbitrary rather than bitwise commutation.
Problem
The paper asks how to partition Pauli operators into simultaneously measurable groups, motivated by reducing the measurements required in each VQE quantum step.
Source: §II · p. 2, §IV · p. 5Method
It reduces Pauli partitioning to graph coloring and compares expected partition sizes for different gate sets and commutation conditions.
Source: §II · p. 2, §III · pp. 2–3Results
The k-partitioning decision problem is NP-complete for k ≥ 3, and the authors conjecture roughly O(m) more parts under bitwise than general commutation.
Source: §II · p. 2, §III · p. 4Takeaways & Limitations
The authors conclude that graph-coloring algorithms should significantly reduce measurements in each VQE quantum step.
Source: §V · p. 5
Abstract
from arXiv · showhide
Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in a multitude of quantum algorithms. Simultaneously measuring sets of operators allows for fewer measurements and an overall speedup of the measurement process. We investigate the task of partitioning a random subset of Pauli operators into simultaneously-measurable parts. Using heuristics from coloring random graphs, we give an upper bound for the expected number of parts in our partition. We go on to conjecture that allowing arbitrary Clifford operators before measurement, rather than single-qubit operations, leads to a decrease in the number of parts which is linear with respect to the lengths of the operators. We give evidence to confirm this conjecture and comment on the importance of this result for a specific near-term application: speeding up the measurement process of the variational quantum eigensolver.
I. INTRODUCTION
The paper develops a generalized Pauli-operator framework motivated by reducing measurement work in the variational quantum eigensolver (VQE). It sets up simultaneous diagonalization and previews the graph-coloring analysis and gate-set comparison that follow.
I. INTRODUCTION
- For prime dimension q, the paper uses shift and clock operators as generalized Pauli operators; at q = 2, they are recognizable as the usual Pauli matrices.The introduction frames the results for prime-dimensional qudits, noting that q can be replaced by 2 for most near-term applications.Source: §I · p. 1
- Simultaneously diagonalizing a group of operators lets the measurement process measure them together in the computational basis.For a gate set GS, DiagGS denotes sets of operators simultaneously diagonalizable by an element of that set.Source: §I · p. 1
- The paper formalizes partitioning arbitrary-length Pauli operators, relates it to graph coloring, compares Clifford gate sets, and examines consequences for VQE measurements.It conjectures that arbitrary Clifford operations reduce the number of parts linearly with operator length and discusses evidence for this in the VQE setting.Source: §I · pp. 1–2
II. PARTITIONING SETS OF PAULI OPERATORS
The section formalizes Pauli partitioning as a decision problem and proves its equivalence to graph coloring. This establishes NP-completeness for the decision version and connects minimum partition size to graph coloring.
II. PARTITIONING SETS OF PAULI OPERATORS
- A set has a k-partition into diagonalizable parts if and only if its corresponding non-diagonalizable graph has a k-coloring.The reduction identifies graph color classes with co-cliques, each representing a diagonalizable set of Pauli operators.Source: §II · p. 2
- kPart is NP-complete for k ≥ 3 with respect to the size of the Pauli set.The reduction uses graphs whose vertex count equals the number of Pauli operators.Source: §II · p. 2
- Finding the fewest diagonalizable parts is exactly equivalent to coloring the corresponding graph with the fewest colors.This equivalence supports analysis of the expected number of parts for randomly chosen Pauli sets.Source: §II · p. 2
III. COMPARING PARTITION SIZES FOR CERTAIN GATE SETS
The section compares expected partition sizes under general and single-qudit Clifford gate sets. It argues that general Clifford operations can reduce the number of parts by roughly O(m), with small examples consistent with that length scaling.
III. COMPARING PARTITION SIZES FOR CERTAIN GATE SETS
- Gate choice also depends on device capabilities: unreliable gates such as entanglement gates may favor more-restrictive sets, while expected partition count is another consideration.Source: §III · p. 2
- For linearly independent S, the minimal partition under C is (log2(q) + o(1)) |S| log2(|S|).Source: §III · p. 3
- For |S| ≤ 2m, the probability of no linear dependence is bounded below by ≈0.288788 at q = 2 and tends toward 1 for larger q.For |S| > 2m, the authors instead heuristically argue that dependence enlarges co-cliques and may lower the number of colors.Source: §III · p. 3
- Under sqC, parts require quditwise-commuting Paulis; the expected count is lower-bounded by m (log2(q) − o(1)) |S| log2(|S|), but dependent edge probabilities block a concrete lower bound.Source: §III · p. 4
- The expected number of parts is roughly O(m) times greater with bitwise than general commutation; graph coloring could improve on greedy partitioning.The comparison follows the authors’ conjectures.Source: §III · p. 4
- In small examples averaging only five sets, the parts ratio grows with O(length), supporting—but not proving—the conjecture.Source: §III · p. 4
IV. MEASUREMENT IN THE VARIATIONAL QUANTUM EIGENSOLVER
The variational quantum eigensolver (VQE) estimates a molecule’s ground-state energy, but its many required runs make reducing measurements important. The authors use greedy graph coloring to partition molecular Hamiltonian Pauli operators under two gate sets and report fewer preparations and measurements with the expanded gate set.
IV. MEASUREMENT IN THE VARIATIONAL QUANTUM EIGENSOLVER
- VQE uses a quantum-classical optimization loop to seek a molecule’s ground-state energy, while estimating the Hamiltonian expectation from its Pauli terms.Expectation values are linear, and each Pauli expectation can be measured by transforming its operator to a diagonal Pauli operator and measuring in the computational basis.Source: §IV · pp. 4–5
- Many near-term devices cannot practically handle the Pauli-operator lengths required as molecule size grows, which determines the number of qubits.The authors identify the sheer number of runs and initializations as a second obstacle, motivating measurement partitioning.Source: §IV · p. 5
- The greedy graph-coloring approach partitions molecular Hamiltonian Pauli operators with respect to generalized Clifford and single-qubit Clifford gate sets.The comparison summarizes results for various molecules.Source: §IV · p. 5
- Changing to the generalized Clifford gate set affords significantly fewer preparations and measurements, and may enable simulation of larger Hamiltonians if entanglement-gate fidelity is ensured.The authors apply greedy graph coloring to molecular Hamiltonians using both generalized and single-qubit Clifford gate sets.Source: §IV · p. 5
V. CONCLUSION AND OPEN PROBLEMS
The paper connects minimal Pauli partitions to graph coloring and states that graph-coloring algorithms should reduce VQE measurements. It also identifies future directions in Hamiltonian diagonalization and application-specific coloring.
V. CONCLUSION AND OPEN PROBLEMS
- Minimal partitions of P^n_q always exist, and the paper links this result to the existence of q^n + 1 mutually unbiased bases in C^(q^n).Source: §V · p. 5
- Pauli partitioning is equivalent in complexity to graph coloring, and implementing graph-coloring algorithms should significantly reduce measurements in each VQE quantum step.Source: §V · p. 5
- A proposed alternative is to diagonalize the Hamiltonian before VQE, though the cited variational quantum state diagonalization algorithm requires more quantum resources than VQE.The authors suggest that a hybrid approach might require fewer quantum resources.Source: §V · p. 5
- Specific coloring algorithms are left open because the molecular Pauli graphs are not truly random; molecular structure or Pauli-term relations could guide application-specific choices.The authors note that optimal colorings might be efficiently calculated in special cases.Source: §V · p. 5
Appendix A: Diagonalization Algorithm
Appendix A gives an inductive construction of a Clifford gate that simultaneously diagonalizes a commuting set of Pauli operators. The construction handles one qudit directly, then reduces the remaining qudits recursively.
Appendix A: Diagonalization Algorithm
- The proof constructs a Clifford gate that simultaneously diagonalizes any commuting set of Pauli operators, and presents the induction as an iterative algorithm.The generalized Clifford gates used in the construction generate the generalized Clifford group.Source: Appendix A · p. 6
- For one qudit, conjugation first removes a Pauli operator’s Z-component and then its X-component, while preserving commutation relations across the set.The base case uses the Fourier gate and a gate that acts on the X-component.Source: Appendix A · p. 6
- For multiple qudits, SUM gates and a single-qudit diagonalizing gate isolate one qudit; the induction hypothesis then diagonalizes the remaining qudits.The construction cancels Z-components, applies Fourier gates to the remaining qudits, and uses commutation to ensure no X-component remains on the isolated qudit.Source: Appendix A · p. 6
- The final Clifford product leaves every operator with no X-component on any qudit, completing the simultaneous diagonalization.The inductive step applies the Clifford gate for the remaining n−1 qudits without affecting the first.Source: Appendix A · p. 6