A central building block of many quantum algorithms is the diagonalization of Pauli operators. Although it is always possible to construct a quantum circuit that simultaneously diagonalizes a given set of commuting Pauli operators, only resource-efficient circuits can be executed reliably on near-term quantum computers. Generic diagonalization circuits, in contrast, often lead to an unaffordable Swap gate overhead on quantum devices with limited hardware connectivity. A common alternative is to exclude two-qubit gates altogether. However, this comes at the severe cost of restricting the class of diagonalizable sets of Pauli operators to tensor product bases (TPBs). In this article, we introduce a theoretical framework for constructing hardware-tailored (HT) diagonalization circuits. Our framework establishes a systematic and highly flexible procedure for tailoring diagonalization circuits with ultra-low gate counts. We highlight promising use cases of our framework and - as a proof-of-principle application - we devise an efficient algorithm for grouping the Pauli operators of a given Hamiltonian into jointly-HT-diagonalizable sets. For several classes of Hamiltonians, we observe that our approach requires fewer measurements than conventional TPB approaches. Finally, we experimentally demonstrate that HT circuits can improve the efficiency of estimating expectation values with cloud-based quantum computers.A central building block of many quantum algorithms is the diagonalization of Pauli operators. Although it is always possible to construct a quantum circuit that simultaneously diagonalizes a given set of commuting Pauli operators, only resource-efficient circuits can be executed reliably on near-term quantum computers. Generic diagonalization circuits, in contrast, often lead to an unaffordable Swap gate overhead on quantum devices with limited hardware connectivity. A common alternative is to exclude two-qubit gates altogether. However, this comes at the severe cost of restricting the class of diagonalizable sets of Pauli operators to tensor product bases (TPBs). In this article, we introduce a theoretical framework for constructing hardware-tailored (HT) diagonalization circuits. Our framework establishes a systematic and highly flexible procedure for tailoring diagonalization circuits with ultra-low gate counts. We highlight promising use cases of our framework and - as a proof-of-principle application - we devise an efficient algorithm for grouping the Pauli operators of a given Hamiltonian into jointly-HT-diagonalizable sets. For several classes of Hamiltonians, we observe that our approach requires fewer measurements than conventional TPB approaches. Finally, we experimentally demonstrate that HT circuits can improve the efficiency of estimating expectation values with cloud-based quantum computers.

arXiv:2203.03646Mar 7, 20224 citations

Reducing circuit depth with qubitwise diagonalization

Edison M. Murairi, Michael J. Cervia

A variety of quantum algorithms employ Pauli operators as a convenient basis for studying the spectrum or evolution of Hamiltonians or measuring multibody observables. One strategy to reduce circuit depth in such algorithms involves simultaneous diagonalization of Pauli operators generating unitary evolution operators or observables of interest. We propose an algorithm yielding quantum circuits with depths $O(nlogr)$ diagonalizing $n$ -qubit operators generated by $r$ Pauli operators. Moreover, as our algorithm iteratively diagonalizes all operators on at least one qubit per step, it is well suited to maintain low circuit depth even on hardware with limited qubit connectivity. We observe that our algorithm performs favorably in producing quantum circuits diagonalizing randomly generated Hamiltonians as well as molecular Hamiltonians with short depths and low two-qubit gate counts.A variety of quantum algorithms employ Pauli operators as a convenient basis for studying the spectrum or evolution of Hamiltonians or measuring multibody observables. One strategy to reduce circuit depth in such algorithms involves simultaneous diagonalization of Pauli operators generating unitary evolution operators or observables of interest. We propose an algorithm yielding quantum circuits with depths $O(nlogr)$ diagonalizing $n$-qubit operators generated by $r$ Pauli operators. Moreover, as our algorithm iteratively diagonalizes all operators on at least one qubit per step, it is well suited to maintain low circuit depth even on hardware with limited qubit connectivity. We observe that our algorithm performs favorably in producing quantum circuits diagonalizing randomly generated Hamiltonians as well as molecular Hamiltonians with short depths and low two-qubit gate counts.

arXiv:2306.00170Physical Review ADec 13, 20232 citations

Optimised Baranyai partitioning of the second quantised Hamiltonian

Bence Csakany, Alex J. W. Thom

Simultaneous measurement of multiple Pauli strings (tensor products of Pauli matrices) is the basis for efficient measurement of observables on quantum computers by partitioning the observable into commuting sets of Pauli strings. We present the implementation and optimisation of the Baranyai grouping method for second quantised Hamiltonian partitioning in molecules up to CH $_4$ (cc-pVDZ, 68 qubits) and efficient construction of the diagonalisation circuit in $O(N)$ quantum gates, compared to $O(N^2)$ , where $N$ is the number of qubits. We show that this method naturally handles sparsity in the Hamiltonian and produces a $O(1)$ number of groups for linearly scaling Hamiltonians, such as those formed by molecules in a line; rising to $O(N^3)$ for fully connected two-body Hamiltonians. While this is more measurements than some other schemes it allows for the flexibility to move Pauli strings and optimise the variance. We also present an explicit optimisation for spin-symmetry which reduces the number of groups by a factor of $8$ , without extra computational effort.Simultaneous measurement of multiple Pauli strings (tensor products of Pauli matrices) is the basis for efficient measurement of observables on quantum computers by partitioning the observable into commuting sets of Pauli strings. We present the implementation and optimisation of the Baranyai grouping method for second quantised Hamiltonian partitioning in molecules up to CH$_4$ (cc-pVDZ, 68 qubits) and efficient construction of the diagonalisation circuit in $O(N)$ quantum gates, compared to $O(N^2)$, where $N$ is the number of qubits. We show that this method naturally handles sparsity in the Hamiltonian and produces a $O(1)$ number of groups for linearly scaling Hamiltonians, such as those formed by molecules in a line; rising to $O(N^3)$ for fully connected two-body Hamiltonians. While this is more measurements than some other schemes it allows for the flexibility to move Pauli strings and optimise the variance. We also present an explicit optimisation for spin-symmetry which reduces the number of groups by a factor of $8$, without extra computational effort.

arXiv:2311.11826Nov 20, 20231 citation

Diagonal operator decomposition on restricted topologies via enumeration of quantum state subsets

Jan Tułowiecki, Łukasz Czerwiński, Konrad Deka +3 more

Various quantum algorithms require usage of arbitrary diagonal operators as subroutines. For their execution on a physical hardware, those operators must be first decomposed into target device's native gateset and its qubit connectivity for entangling gates. Here, we assume that the allowed gates are exactly the CX gate and the parameterized phase gate. We introduce a framework for the analysis of CX-only circuits and through its lens provide solution constructions for several different device topologies (fully-connected, linear and circular). We also introduce two additional variants of the problem. Those variants can be used in place of exact decomposition of the diagonal operator when the circuit following it satisfies a set of prerequisites, enabling further reduction in the CX cost of implementation. Finally, we discuss how to exploit the framework for the decomposition of a particular, rather than general, diagonal operator.Various quantum algorithms require usage of arbitrary diagonal operators as subroutines. For their execution on a physical hardware, those operators must be first decomposed into target device's native gateset and its qubit connectivity for entangling gates. Here, we assume that the allowed gates are exactly the CX gate and the parameterized phase gate. We introduce a framework for the analysis of CX-only circuits and through its lens provide solution constructions for several different device topologies (fully-connected, linear and circular). We also introduce two additional variants of the problem. Those variants can be used in place of exact decomposition of the diagonal operator when the circuit following it satisfies a set of prerequisites, enabling further reduction in the CX cost of implementation. Finally, we discuss how to exploit the framework for the decomposition of a particular, rather than general, diagonal operator.

arXiv:2403.02109Mar 4, 20240 citations

Hardware-Friendly Estimation of LCU-Based Expectation Values via Quantum-to-Classical Workload Shifting

Hari Hara Suthan Chittoor, Thirumalai Manimaran

Estimating the expectation value of an observable $O$ for a quantum state evolved under a general, possibly non-unitary, operator $A$ is a fundamental task in quantum simulation, chemistry, and variational algorithms. Existing linear-combination-of-unitaries (LCU) approaches implement $A$ on hardware, requiring ancilla qubits, controlled unitaries and, in randomized variants, fresh circuit executions for every Monte Carlo sample. These costs are prohibitive on NISQ and early fault-tolerant devices. We propose HERMES (Hardware-Efficient Reusable Measurements for Estimating LCU-based expectation valueS), which moves the processing of $A$ entirely to the classical domain. Expanding $A$ and $O$ in the Pauli basis and sampling pairs of Pauli terms classically, HERMES rewrites $μ= \mathrm{Tr}(OAρA^{\dagger})/\mathrm{Tr}(AρA^{\dagger})$ as a ratio of weighted sums of Pauli expectation values on the unmodified state $ρ$ , with no ancillas, controlled unitaries, or quantum implementation of $A$ . A two-level measurement reduction scheme (Pauli deduplication and commuting-group partitioning) bounds the number of distinct measurement settings independently of the sample count, unlike randomized LCU methods whose circuit executions grow linearly with it. On ground-state energy estimation of $\mathrm{H}_2$ and $\mathrm{LiH}$ via imaginary-time evolution, HERMES matches or beats Pauli single-ancilla LCU baseline using shallow circuits with no two-qubit gates beyond state preparation under qubit-wise commuting grouping. It is also more robust to noise and completes a noisy $\mathrm{LiH}$ simulation infeasible for the baseline, at the cost of classical overhead growing with the number of Pauli terms in $A$ .Estimating the expectation value of an observable $O$ for a quantum state evolved under a general, possibly non-unitary, operator $A$ is a fundamental task in quantum simulation, chemistry, and variational algorithms. Existing linear-combination-of-unitaries (LCU) approaches implement $A$ on hardware, requiring ancilla qubits, controlled unitaries and, in randomized variants, fresh circuit executions for every Monte Carlo sample. These costs are prohibitive on NISQ and early fault-tolerant devices. We propose HERMES (Hardware-Efficient Reusable Measurements for Estimating LCU-based expectation valueS), which moves the processing of $A$ entirely to the classical domain. Expanding $A$ and $O$ in the Pauli basis and sampling pairs of Pauli terms classically, HERMES rewrites $μ= \mathrm{Tr}(OAρA^{\dagger})/\mathrm{Tr}(AρA^{\dagger})$ as a ratio of weighted sums of Pauli expectation values on the unmodified state $ρ$, with no ancillas, controlled unitaries, or quantum implementation of $A$. A two-level measurement reduction scheme (Pauli deduplication and commuting-group partitioning) bounds the number of distinct measurement settings independently of the sample count, unlike randomized LCU methods whose circuit executions grow linearly with it. On ground-state energy estimation of $\mathrm{H}_2$ and $\mathrm{LiH}$ via imaginary-time evolution, HERMES matches or beats Pauli single-ancilla LCU baseline using shallow circuits with no two-qubit gates beyond state preparation under qubit-wise commuting grouping. It is also more robust to noise and completes a noisy $\mathrm{LiH}$ simulation infeasible for the baseline, at the cost of classical overhead growing with the number of Pauli terms in $A$.

quant-pharXiv:2610.32841Oct 26, 20260 citations

Circuit optimization of Hamiltonian simulation by simultaneous diagonalization of Pauli clusters

Ewout van den Berg, Kristan Temme

Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitioning sets of Paulis into commuting subsets, and show that the proposed approach can help to significantly reduce both the number of C N O T operations and circuit depth for Hamiltonians arising in quantum chemistry. The algorithms for simultaneous diagonalization are also applicable in the context of stabilizer states; in particular we provide novel four- and five-stage representations, each containing only a single stage of conditional gates.Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitioning sets of Paulis into commuting subsets, and show that the proposed approach can help to significantly reduce both the number of C N O T operations and circuit depth for Hamiltonians arising in quantum chemistry. The algorithms for simultaneous diagonalization are also applicable in the context of stabilizer states; in particular we provide novel four- and five-stage representations, each containing only a single stage of conditional gates.

arXiv:2003.13599QuantumSep 12, 202053 citations

O(nlogr)

diagonalizing

n

-qubit operators generated by

r

Pauli operators. Moreover, as our algorithm iteratively diagonalizes all operators on at least one qubit per step, it is well suited to maintain low circuit depth even on hardware with limited qubit connectivity. We observe that our algorithm performs favorably in producing quantum circuits diagonalizing randomly generated Hamiltonians as well as molecular Hamiltonians with short depths and low two-qubit gate counts.

_4

(cc-pVDZ, 68 qubits) and efficient construction of the diagonalisation circuit in

O(N)

quantum gates, compared to

O(N^2)

, where

N

is the number of qubits. We show that this method naturally handles sparsity in the Hamiltonian and produces a

O(1)

number of groups for linearly scaling Hamiltonians, such as those formed by molecules in a line; rising to

O(N^3)

for fully connected two-body Hamiltonians. While this is more measurements than some other schemes it allows for the flexibility to move Pauli strings and optimise the variance. We also present an explicit optimisation for spin-symmetry which reduces the number of groups by a factor of

8

, without extra computational effort.

Estimating the expectation value of an observable

O

for a quantum state evolved under a general, possibly non-unitary, operator

A

is a fundamental task in quantum simulation, chemistry, and variational algorithms. Existing linear-combination-of-unitaries (LCU) approaches implement

A

on hardware, requiring ancilla qubits, controlled unitaries and, in randomized variants, fresh circuit executions for every Monte Carlo sample. These costs are prohibitive on NISQ and early fault-tolerant devices. We propose HERMES (Hardware-Efficient Reusable Measurements for Estimating LCU-based expectation valueS), which moves the processing of

A

entirely to the classical domain. Expanding

A

and

O

in the Pauli basis and sampling pairs of Pauli terms classically, HERMES rewrites

μ= \mathrm{Tr}(OAρA^{\dagger})/\mathrm{Tr}(AρA^{\dagger})

as a ratio of weighted sums of Pauli expectation values on the unmodified state

ρ

, with no ancillas, controlled unitaries, or quantum implementation of

A

. A two-level measurement reduction scheme (Pauli deduplication and commuting-group partitioning) bounds the number of distinct measurement settings independently of the sample count, unlike randomized LCU methods whose circuit executions grow linearly with it. On ground-state energy estimation of

\mathrm{H}_2

and

\mathrm{LiH}

via imaginary-time evolution, HERMES matches or beats Pauli single-ancilla LCU baseline using shallow circuits with no two-qubit gates beyond state preparation under qubit-wise commuting grouping. It is also more robust to noise and completes a noisy

\mathrm{LiH}

simulation infeasible for the baseline, at the cost of classical overhead growing with the number of Pauli terms in

A