Quantum gate decompositions of many-particle state preparations for simulations of quantum chemistry

Bryan Gard, Linghua Zhu, George S. Barron +3 more

Abstract The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediate-scale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multi-qubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that respect particle number, total spin, spin projection, and time-reversal symmetries. These circuits contain the minimal number of variational parameters needed to fully span the appropriate symmetry subspace dictated by the chemistry problem while avoiding all irrelevant sectors of Hilbert space. We show how to construct these circuits for arbitrary numbers of orbitals, electrons, and spin quantum numbers, and we provide explicit decompositions and gate counts in terms of standard gate sets in each case. We test our circuits in quantum simulations of the $${H}_{2}$$ H 2 and $$LiH$$ L i H molecules and find that they outperform standard state preparation methods in terms of both accuracy and circuit depth.

Viktor Khinevich, Wataru Mizukami

We present systematic and resource-efficient constructions of continuous symmetry projectors, particularly $U(1)$ particle number and $SU(2)$ total spin, tailored for fault-tolerant quantum computations. Our approach employs a linear combination of unitaries (LCU) as well as generalized quantum signal processing (GQSP and GQSVT) to implement projectors. These projectors can then be coherently applied as state filters prior to quantum phase estimation (QPE). We analyze their asymptotic gate complexities for explicit circuit realizations. For the particle number and $S_z$ symmetries, GQSP offers favorable resource usage features owing to its low ancilla qubit requirements and robustness to finite precision rotation gate synthesis. For the total spin projection, the structured decomposition of $\hat{P}_{S,M_S}$ reduces the projector T gate count. Numerical simulations show that symmetry filtering substantially increases the QPE success probability, leading to a lower overall cost compared to that of unfiltered approaches across representative molecular systems. Resource estimates further indicate that the cost of symmetry filtering is $3$ to $4$ orders of magnitude lower than that of the subsequent phase estimation step This advantage is especially relevant in large, strongly correlated systems, such as FeMoco, a standard strongly correlated open-shell benchmark. For FeMoco, the QPE cost is estimated at ${\sim}10^{10}$ T gates, while our symmetry projector requires only ${\sim}10^{6}$--$10^{7}$ T gates. These results establish continuous-symmetry projectors as practical and scalable tools for state preparation in quantum chemistry and provide a pathway toward realizing more efficient fault-tolerant quantum simulations.

Smriti Sharma

Nikos L. Doltsinis

Baptiste Claudon, Alexis Lucas, Jean-Philip Piquemal +2 more

Quantum state preparation is a central primitive in many quantum algorithms, yet it is generally resource intensive, with efficient constructions known only for structured families of states. This work introduces a method for preparing quantum states whose amplitudes are given by a degree$-d$ polynomial, using circuits with logarithmic depth in the number $n$ of qubits and only $\mathcal O(n)$ ancilla qubits, improving previous approaches that required linear-depth circuits. The construction first relies on a block-encoding of an affine diagonal operator based on its Pauli-basis decomposition, which involves only $n$ terms. A modified linear-combination-of-unitaries (LCU) technique is introduced to implement this decomposition in logarithmic depth, together with a novel circuit for the EXACT-one oracle that flags basis states in which exactly one qubit is in the state $|1\rangle$. It then uses a generalized quantum eigenvalue transformation (GQET) to promote this affine operator to an arbitrary degree polynomial. Theoretical analysis and numerical simulations are reported along with a proof-of-principle implementation on a trapped-ion quantum processor using $14$ qubits and more than $500$ primitive quantum gates. Because polynomial approximations are ubiquitous in scientific computing, this construction provides a scalable and resource-efficient approach to quantum state preparation, further improving the potential of quantum algorithms in fields such as chemistry, physics, engineering, and finance.

Juan Miguel Arrazola, Olivia Di Matteo, Nicolás Quesada +3 more

Universal gate sets for quantum computing have been known for decades, yet no universal gate set has been proposed for particle-conserving unitaries, which are the operations of interest in quantum chemistry. In this work, we show that controlled single-excitation gates in the form of Givens rotations are universal for particle-conserving unitaries. Single-excitation gates describe an arbitrary U ( 2 ) rotation on the two-qubit subspace spanned by the states | 01 ⟩ , | 10 ⟩ , while leaving other states unchanged – a transformation that is analogous to a single-qubit rotation on a dual-rail qubit. The proof is constructive, so our result also provides an explicit method for compiling arbitrary particle-conserving unitaries. Additionally, we describe a method for using controlled single-excitation gates to prepare an arbitrary state of a fixed number of particles. We derive analytical gradient formulas for Givens rotations as well as decompositions into single-qubit and CNOT gates. Our results offer a unifying framework for quantum computational chemistry where every algorithm is a unique recipe built from the same universal ingredients: Givens rotations.