Hamiltonian simulation: product formulas, qubitization and QSVT
Algorithms and benchmarks / Simulation of physical systems
Reproduces how a quantum system evolves in time on a gate-based quantum computer: spins, electrons in materials, molecules or nuclear matter. This is the application Feynman had in mind, and the one with the strongest case for exponential speedups.
- Era
- Fault-tolerant
- Problem class
- Simulation of physical systems
- Speedup
- Exponential for many physical models compared with exact classical simulation; qubitization needs O(t + log(1/epsilon)) queries for time t and error epsilon (Low and Chuang 2019).
How it works
Product (Trotter) formulas split the evolution into short steps of simple terms. Qubitization embeds the Hamiltonian in a block of a larger unitary and reaches the optimal query cost; quantum singular value transformation (QSVT) unifies these methods and many other algorithms.
Typical use cases
- Dynamics of magnets and materials (condensed matter).
- Fermi-Hubbard model and electronic structure as a step towards energies.
Documented use cases on related problems
From the use case directory; the sources of these cases may use other methods than this one.
Resource estimates
Figures exactly as stated by their authors, with their assumptions. They are estimates under those assumptions, not forecasts, and they tend to fall as methods improve.
| Quantity | Value as stated |
|---|---|
| Physical qubits | about a million superconducting qubits |
| Runtime | a matter of hours |
Assumptions: Circuits compiled to surface-code fault-tolerant gates with a per-gate error rate of one part in a thousand (10^-3); T complexity linear in the number of orbitals.
Source: R. Babbush et al. 2018, Phys. Rev. X 8, 041015 (2018); arXiv:1805.03662
Demonstrations
- 2024: IBM’s 127-qubit kicked Ising experiment on its Eagle processor was later reproduced classically with a tensor network more accurately and precisely than the quantum processor did. (J. Tindall et al. 2024)
Hardware platforms that fit
- Superconducting qubits: The cited estimate for classically intractable Hubbard and electronic-structure instances assumes about a million superconducting qubits with surface-code gates (Babbush et al. 2018); a 127-qubit kicked Ising experiment ran on IBM’s Eagle processor (Tindall et al. 2024).
What you need to run it
- Classical simulation
- A state-vector simulator holds 2^n complex amplitudes, so memory doubles with every qubit: about 30 qubits fit in 16 GiB at double precision. Circuits with little entanglement can go further with tensor-network methods. See the guide for the calculator. Tensor networks follow the entanglement, which grows with simulated time.
- Noisy hardware (NISQ)
- Short-time dynamics of up to about a hundred qubits with error mitigation; check results against classical tensor-network simulation.
- Fault-tolerant machine
- For the Babbush et al. (2018) instances, about a million physical superconducting qubits at 10^-3 gate error for a few hours.
- Annealer or analogue simulator
- Analogue simulators reproduce some models directly; see analogue quantum simulation.
Calculate the memory of a simulation
Where to run it in Europe
Superconducting qubits
- VLQ (Czechia; Inaugurated, per the EuroHPC JU): Superconducting; digital quantum computer. Access: EuroHPC JU call for quantum access in pilot mode. Deployment milestones VLQ
- Euro-Q-Exa (Germany; Inaugurated, per the EuroHPC JU): Superconducting; digital quantum computer. Access: EuroHPC JU call for quantum access in pilot mode. Deployment milestones Euro-Q-Exa
- EuroQCS-Spain (Spain; Inaugurated, per the EuroHPC JU): Superconducting annealing; analogue quantum computer. Access: EuroHPC JU call for quantum access in pilot mode. Deployment milestones EuroQCS-Spain
- Quantum Inspire (Netherlands; In operation, per the source): Online platform with superconducting transmon QPUs, a neutral-atom emulator and simulators; a Spin-2+ electron spin QPU is listed as archived. Access: Quantum Inspire registration.
- LRZ Quantum Integration Centre (QIC) (Germany; In operation, per the source): Superconducting systems from IQM (54 qubits, up to 150 in future), a 20-qubit AQT ion-trap system, and a neutral-atom system being developed by planqc (MAQCS project). Access: LRZ quantum computing consulting.
- Finnish Quantum-Computing Infrastructure (FiQCI) (Finland; In operation, per the source): Hybrid HPC and quantum computing platform: VTT Q50 (53 qubits) and Aalto Q20 (20 qubits), superconducting systems made by IQM, used through the LUMI supercomputer environment. Access: FiQCI calls for academic projects.
- Quantum Spain quantum computer at BSC-CNS (Spain; In operation, per the source): Superconducting; digital quantum computer with a 35-qubit processor, integrated into the MareNostrum 5 supercomputer. Access: Spanish Supercomputing Network (RES) quantum access. Deployment milestones Quantum Spain quantum computer at BSC-CNS
The full infrastructure catalogue · The infrastructure deployment tracker
Software
European tools in the software directory
Other open-source frameworks
- OpenFermion (Google Quantum AI (United States); non-European)
- Qiskit (IBM (United States); non-European)
A listing is not an endorsement; check each tool for the features you need.
Limitations and open questions
- Noisy devices reach only short times; classical tensor-network methods have matched or beaten large noisy experiments (Tindall et al. 2024).
- Childs et al. (2018) find circuits for spin systems orders of magnitude smaller than for factoring or chemistry, yet still beyond noisy machines.
In the Quantum Algorithm Zoo
- Simulating Quantum Hamiltonian Dynamics (Quantum Algorithm Zoo, external site): Approximation and Simulation Algorithms; speedup as the Zoo states it: Superpolynomial
Index: the Quantum Algorithm Zoo by Stephen P. Jordan, last updated 2026-09-10, checked 2026-10-23. The portal links to it and copies no description from it. Browse the full Quantum Algorithm Zoo (external site)
Named in European procurements
- Used in: EuroHPC procurement LUMI-Q (now VLQ) (Required in the benchmark suite (scored under the award criteria))
As the tender documents state; how a benchmark is weighted against others is described in each procurement.
Sources
- A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, Y. Su (2018). Toward the first quantum simulation with quantum speedup. PNAS 115, 9456 (2018); arXiv:1711.10980. Read: abstract page, retrieved 2026-10-23.
- G. H. Low, I. L. Chuang (2019). Hamiltonian Simulation by Qubitization. Quantum 3, 163 (2019); arXiv:1610.06546. Read: abstract page, retrieved 2026-10-23.
- A. Gilyén, Y. Su, G. H. Low, N. Wiebe (2019). Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. Proceedings of STOC 2019, 193; arXiv:1806.01838. Read: abstract page, retrieved 2026-10-23.
- R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, H. Neven (2018). Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity. Phys. Rev. X 8, 041015 (2018); arXiv:1805.03662. Read: abstract page, retrieved 2026-10-23.
- J. Tindall, M. Fishman, M. Stoudenmire, D. Sels (2024). Efficient tensor network simulation of IBM's Eagle kicked Ising experiment. PRX Quantum 5, 010308 (2024); arXiv:2306.14887. Read: abstract page, retrieved 2026-10-23.