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What you need to run quantum algorithms

Algorithms and benchmarks

What it takes to run a quantum algorithm, in four situations: simulating it on classical computers, running it on noisy hardware today, running it on a future fault-tolerant machine, and using an annealer or analogue simulator. The last part lists where in Europe each kind of hardware can be used.

Educational guidance only: no procurement or investment advice. Hardware figures are dated and sourced; providers publish newer ones.

Classical simulation

A simulator that stores the full quantum state keeps 2^n complex amplitudes for n qubits. In double precision (complex128, two 64-bit floats) each amplitude takes 16 bytes, so the memory is 2^n × 16 bytes; in single precision (complex64, two 32-bit floats) 8 bytes. A density matrix, needed to simulate noise exactly, has 4^n entries. Each extra qubit doubles the memory of a state vector. (NumPy developers 2026)

Memory of a state vector by number of qubits
Qubitscomplex128 (binary units)complex128 (decimal units)complex64
2016 MiB16.8 MB8 MiB
3016 GiB17.2 GB8 GiB
4016 TiB17.6 TB8 TiB
5016 PiB18 PB8 PiB

Which computer fits

  • Laptop (16 GiB): up to 30 qubits at double precision (state vector only).
  • Workstation (256 GiB): up to 34 qubits at double precision (state vector only).
  • One large HPC node (1 TiB): up to 36 qubits at double precision (state vector only).
  • Large partition of a supercomputer (1 PiB, distributed): up to 46 qubits at double precision (state vector only).

Illustrative orders of magnitude chosen by the portal, not the memory of a named system. Simulators also need working memory and fast networks between nodes; check the documentation of the system you use.

Beyond state vectors

Tensor-network and matrix product state (MPS) simulators store the state in pieces whose size depends on entanglement, not only on the number of qubits: the cost grows linearly with the qubits and exponentially with the entanglement. They simulate wide but shallow or weakly entangled circuits far beyond 50 qubits, and they fail on deep, highly entangled ones. Stabiliser simulators handle Clifford circuits, such as error-correction circuits, with thousands of qubits. (G. Vidal 2003)

Documented large simulations

  • Random circuits of up to 38 qubits simulated by QuEST on up to 2048 compute nodes of the ARCUS Phase-B and ARCHER supercomputers. (T. Jones et al. 2019)
  • The Jülich universal quantum computer simulator ran 48-qubit simulations on the Sunway TaihuLight and K computer; adaptive encoding of the wave function cut its memory need by a factor of eight. (H. De Raedt et al. 2019)
  • Tensor-network sampling of the 53-qubit, 20-cycle Sycamore circuits at low fidelity in about 15 hours on 512 GPUs. (F. Pan et al. 2022)
  • Tensor-network simulation of a 127-qubit kicked Ising experiment, more accurate than the quantum processor. (J. Tindall et al. 2024)

State-vector memory calculator

Enter a number of qubits and press Calculate; the page works without JavaScript.

Noisy hardware (NISQ)

On noisy hardware the useful size of a circuit is set less by the number of qubits than by the error of each two-qubit gate: a circuit with thousands of two-qubit gates needs errors well below one in a thousand to give a usable signal. Other needs: connectivity that matches the problem (or extra swap gates), coherence times long compared with the circuit, mid-circuit measurement and reset for some algorithms, and many shots, often thousands per estimated value, so throughput matters. Error mitigation trades more shots for less bias.

Dated figures from the sources, by platform (the newest machines may do better):

Superconducting qubits
Trapped ions
Neutral atoms
Photonic qubits
Spin qubits in semiconductors
  • Single- and two-qubit gate fidelities (silicon, gate set tomography): all above 99.5% (X. Xue et al. 2022)

How such figures are measured: the benchmarks

Fault-tolerant machines

Fault-tolerant algorithms run on logical qubits, each encoded in many physical qubits by an error-correcting code such as the surface code. The code distance sets how strongly errors are suppressed and how many physical qubits each logical qubit needs; physical errors must stay below the threshold of the code. Gates such as T or Toffoli need magic states from dedicated factories, which is why estimates count them. Results are quoted for a stated physical error rate, cycle time and connectivity.

Published resource estimates (assumptions on each algorithm page)
TaskFigures as statedSource
Factoring a 2048-bit RSA integerNoisy physical qubits: 20 million; Runtime: 8 hours; Logical qubits (formula for an n-bit integer): 3n + 0.002 n lg n(C. Gidney et al. 2021)
Factoring a 2048-bit RSA integerNoisy physical qubits: fewer than one million; Runtime: less than a week(C. Gidney 2025)
Factoring a 2048-bit RSA integer with a multimode memoryPhysical qubits in the processor: 13,436; Memory: 28 million spatial modes and 45 temporal modes; Runtime: 177 days(É. Gouzien et al. 2021)
256-bit elliptic curve discrete logarithm with cat qubitsCat qubits: 126,133; Runtime: 9 hours(É. Gouzien et al. 2023)
256-bit elliptic curve discrete logarithm (NIST P-256), logical levelLogical qubits (upper bound): 2,330; Toffoli gates (upper bound): about 1.29 × 10^11(M. Roetteler et al. 2017)
256-bit elliptic curve private key, logical levelToffoli gates per key: about 50 million(D. Litinski 2023)
Pricing autocallable and TARF derivatives with an advantage over classical Monte CarloLogical qubits: 8k; T-depth: 54 million; Time needed for advantage: execution in the order of a second(S. Chakrabarti et al. 2021)
Classically intractable instances of the Hubbard model and of electronic structurePhysical qubits: about a million superconducting qubits; Runtime: a matter of hours(R. Babbush et al. 2018)
Ground-state energy of FeMocoPhysical qubits: about four million; Runtime: under four days(J. Lee et al. 2021)
A fault-tolerant memory of 12 logical qubitsPhysical qubits: 288; Syndrome cycles preserved: nearly one million; Surface code equivalent: about 3,000 physical qubits(S. Bravyi et al. 2024)
256-bit elliptic curve discrete logarithmCat qubits: 126,133; Runtime: 9 hours(É. Gouzien et al. 2023)

Error correction is being demonstrated now: see the surface code, qLDPC codes and cat qubits. Surface code demonstrations

Annealers and analogue simulators

Annealers solve optimisation problems written as Ising or QUBO models: the problem graph must be embedded in the hardware graph, which uses several physical qubits per variable when couplings are missing. Analogue simulators reproduce a model directly; what matters is the number of atoms or ions, the geometry and interactions they can program, how well they are calibrated and whether results can be checked classically.

Quantum annealing
Analogue quantum simulators

Quantum annealing · Analogue quantum simulation

Where to run it in Europe

Quantum computers, simulators and user platforms of the infrastructure catalogue, by the technology the catalogue records, with the access route it names. Capacities and access conditions are those of the catalogue and its sources; check them with the provider.

Superconducting qubits

Trapped ions

Neutral atoms

Photonic qubits

Spin qubits in semiconductors

Quantum annealing

Analogue quantum simulators

The full infrastructure catalogue · The infrastructure deployment tracker

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