Quantum phase estimation
Algorithms and benchmarks / Linear algebra and eigenvalues
Measures an eigenvalue of a unitary operation, such as the energy of a molecule encoded as a phase. It is the core subroutine of Shor’s algorithm, of fault-tolerant chemistry and of many linear-algebra algorithms.
- Era
- Fault-tolerant
- Problem class
- Linear algebra and eigenvalues
- Speedup
- Precision epsilon with about 1/epsilon applications of the unitary (Heisenberg-limited), against about 1/epsilon squared for sampling-based estimates.
How it works
Control qubits steer powers of the unitary acting on an eigenstate; an inverse quantum Fourier transform turns the accumulated phases into a binary readout of the eigenvalue (Kitaev 1995). Newer variants trade the many control qubits for one ancilla and classical post-processing.
Typical use cases
- Ground-state energies in chemistry and materials, through Hamiltonian simulation.
- A building block of factoring and of linear-system solvers.
Resource estimates
No resource estimate with stated assumptions was verified for this family in this release.
Demonstrations
- 2016: Energy surface of molecular hydrogen with Trotterised phase estimation and with VQE on a programmable array of superconducting qubits. (P. J. J. O'Malley et al. 2016)
Hardware platforms that fit
- Superconducting qubits: The energy surface of molecular hydrogen was computed on superconducting qubits with Trotterisation and phase estimation as well as with VQE (O’Malley et al. 2016).
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.
- Noisy hardware (NISQ)
- Only small demonstrations; single-ancilla variants (Lin and Tong 2022) aim at early fault-tolerant machines.
- Fault-tolerant machine
- The main workload of fault-tolerant chemistry; see the resource estimates of electronic structure.
- Annealer or analogue simulator
- Not applicable for this family.
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
A listing is not an endorsement; check each tool for the features you need.
Limitations and open questions
- Textbook phase estimation needs many ancilla qubits, deep circuits and an inverse quantum Fourier transform, which is costly on early fault-tolerant hardware (Lin and Tong 2022).
- It needs an initial state with a good overlap with the wanted eigenstate; preparing one can be hard.
In the Quantum Algorithm Zoo
No entry of its own in the Zoo; it appears as a subroutine within other entries.
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))
- Used in: EuroHPC procurement MeluXina-Q (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. Yu. Kitaev (1995). Quantum measurements and the Abelian Stabilizer Problem. arXiv:quant-ph/9511026 (1995). Read: abstract page, retrieved 2026-10-23.
- L. Lin, Y. Tong (2022). Heisenberg-limited ground state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022); arXiv:2102.11340. Read: abstract page, retrieved 2026-10-23.
- P. J. J. O'Malley et al. (2016). Scalable Quantum Simulation of Molecular Energies. Phys. Rev. X 6, 031007 (2016); arXiv:1512.06860. Read: abstract page, retrieved 2026-10-23.