In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous best techniques for industrially and biologically relevant molecules, including the nitrogenase cofactor (FeMoCo) and cytochrome P450. Benchmark calculations comparing our method with the original double-factorization method show a 75% reduction in the block-encoding scaling constant. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the journey to the feasibility of practical Hamiltonian simulations.
Quantum phase estimation based on qubitization is the state-of-the-art fault-tolerant quantum algorithm for computing ground-state energies in chemical applications. In this context, the 1-norm of the Hamiltonian plays a fundamental role in determining the total number of required iterations and also the overall computational cost. In this work, we introduce the symmetry-compressed double factorization (SCDF) approach, which combines a CDF of the Hamiltonian with the symmetry shift technique, significantly reducing the 1-norm value. The effectiveness of this approach is demonstrated numerically by considering various benchmark systems, including the FeMoco molecule, cytochrome P450, and hydrogen chains of different sizes. To compare the efficiency of SCDF to other methods in absolute terms, we estimate Toffoli gate requirements, which dominate the execution time on fault-tolerant quantum computers. For the systems considered here, SCDF leads to a sizable reduction of the Toffoli gate count in comparison to other variants of DF or even tensor hypercontraction, which is usually regarded as the most efficient approach for qubitization.
Simulation of molecular vibrations is emerging as a promising application of quantum computing in quantum chemistry. In this work, we develop new strategies for block-encoding the second-quantized
n-mode representation of the vibrational Hamiltonian, whose algebraic structure closely parallels that of the second-quantized electronic Hamiltonian. We derive linear-combination-of-unitaries (LCU) representations of this Hamiltonian based on Pauli decomposition and tensor hypercontraction (THC) and construct efficient circuits for the corresponding block encodings. We also extend the Block-Invariant Symmetry Shift (BLISS) technique to vibrational Hamiltonians, yielding a lower LCU 1-norm and, consequently, a lower overall circuit cost. For the molecules considered here, BLISS-THC reduces the Toffoli cost of quantum phase estimation by more than two orders of magnitude relative to existing block-encoding approaches. In comparison with recent Trotter-based methods, it shows up to a ninefold reduction in the T-gate count, with the relative savings tending to increase with system size. Asymptotically, and up to polylogarithmic factors, both the Pauli and THC LCUs trade additional ancillas for a quadratic improvement in scaling with the number of modes, while THC provides an additional quadratic improvement in scaling with the number of modals. These results highlight the importance of Hamiltonian factorizations and Toffoli-qubit tradeoffs for reducing the computational cost of quantum simulations of molecular vibrations.
Fault tolerant quantum simulation via the phase estimation algorithm and qubitization has a T-gate count that scales proportionally to the 1-norm of the Hamiltonian, the cost of block encoding the Hamiltonian, and inversely proportionally to the desired accuracy. Tensor factorization methods have been successfully used to reduce T-gate counts in the ground state electronic structure problem. Here we introduce the use of tensor factorization methods to reduce the T-gate count of quantum phase estimation. In particular, we show how Canonical Polyadic and Tucker decompositions of the tensors representing the vibrational and vibronic Hamiltonians can be utilized to rewrite the Hamiltonian in terms of linear combination of bosonic position operators representing nuclear vibrations. We demonstrate the use of these factorization methods on the water and monodeutered methane molecules.
Computational cost of energy estimation for molecular electronic Hamiltonians via quantum phase estimation (QPE) grows with the difference between the largest and smallest eigenvalues of the Hamiltonian. In this work, we propose a preprocessing procedure that reduces the norm of the Hamiltonian without changing its eigenspectrum for the target states of a particular symmetry. The new procedure, block-invariant symmetry shift (BLISS), builds an operator T̂ such that the cost of implementing H ^ − T ^ is reduced compared to that of Ĥ, yet H ^ − T ^ acts on the subspaces of interest the same way as Ĥ does. BLISS performance is demonstrated for a linear combination of unitaries (LCU)-based QPE approaches on a set of small molecules. Using the number of electrons as the symmetry specifying the target set of states, BLISS provided a factor of 2 reduction of 1-norm for several LCU decompositions compared to their unshifted versions.
Electronic structure calculations of molecular systems are among the most promising applications for fault-tolerant quantum computing (FTQC) in quantum chemistry and drug design. However, while recent algorithmic advancements such as qubitization and tensor hypercontraction (THC) have significantly reduced the complexity of such calculations, they do not yet achieve computational runtimes short enough to be practical for industrially relevant use cases. In this work, we combine several advances to electronic structure calculation for molecular systems, resulting in a 2-orders-of-magnitude speedup of estimated runtimes over prior-art algorithms run on comparable quantum devices. One of these advances is a novel framework for block-invariant symmetry-shifted tensor hypercontraction, with which we achieve the tightest Hamiltonian factorizations reported to date. We compile our algorithm for an active volume (AV) architecture, a technical layout that has recently been proposed for fusion-based photonic quantum hardware. AV compilation contributes towards a lower runtime of our computation by eliminating overheads stemming from connectivity issues in the underlying surface code. We present a detailed benchmark of our approach, focusing primarily on the computationally challenging benchmark molecule P450. Leveraging a number of hardware trade-offs in interleaving-based photonic FTQC, we estimate runtimes for the electronic structure calculation of P450 as a function of the device footprint.
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
Sz 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
P^S,MS 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
∼1010 T gates, while our symmetry projector requires only
∼106--
107 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.