Quantum computing for whole-cell modelling: a perspective

Simulating a whole cell, meaning every functional system across a full life cycle, is still out of reach. This Perspective, written with colleagues at Oak Ridge National Laboratory, asks a narrower question than whether quantum computing can help biology. It asks which specific computational kernels inside whole-cell modelling have scaling that a quantum algorithm could improve, and what would have to be true of the hardware before that improvement is real. The answer is narrow. The accounting favours quantum only once spatial discretisation makes the system very large, and it points to a classical HPC solver with a QPU attached rather than to a quantum whole-cell simulator.

The cost of one E. coli trajectory, made concrete

Even the simplest living cell holds thousands of interacting proteins and metabolites, on the order of trillions of atoms. Their functional dynamics span roughly five orders of magnitude in space, from nanometres to micrometres, and nearly nineteen in time, from femtoseconds to hours. Many of the governing physical and chemical properties are not characterised well enough to write down in the first place.

Take a stochastic spatial simulation of one E. coli cell cycle under the reaction–diffusion master equation. A 1 µm³ cell is discretised into 10⁶ voxels of 10 nm edge, with 4,000 species and about 2.4 × 10⁴ reaction channels per voxel once diffusion across the six voxel faces is counted. That comes to roughly 2.9 × 10¹⁶ operations for a single trajectory. On one optimised workstation core that is about 92 years. On Frontier with ideal domain decomposition it is 1.3 hours, and with realistic inter-voxel communication overhead it is 13 hours to 5.5 days. That is one trajectory and one cell cycle, and stochastic answers need many trajectories.

Quantum computing is often invoked for biology, rarely with specifics. This paper names the kernels, puts complexity bounds next to them, and states where the accounting does not favour quantum at all.

The hardware evidence runs opposite to problem size. The smallest scale has a protein-scale demonstration, and the largest has no biological demonstration at all.

Scale	Classical	Quantum candidates	Hardware status
Atomistic–molecular (10⁻⁹–10⁻⁶ m, fs–ms)	MD with force fields; DFT, coupled-cluster, QM/MM	VQE, QAOA, QPE, SQD, Hamiltonian simulation	Furthest along; protein-scale demo exists
Networks (10⁻⁶ m, ms–ks)	FBA (LP/QP), Boolean and probabilistic regulatory networks, Gillespie SSA	Quantum convex optimisation, Grover search and QAOA on QUBO forms, quantum Monte Carlo	Five-node regulatory logic on gate-based hardware
Whole-cell spatial (10⁻⁵–10⁻⁴ m, ms–ks)	Reaction–diffusion PDEs by FEM/FDM; RDME, particle-based	QLSA, Schrödingerisation, Carleman linearisation, tensor networks, quantum Markov chains and walks	No biological demo; PDE and linear-system kernels only

At the atomistic scale the target is accuracy, not speed

DFT approximations carry functional-dependent errors for transition metals, spin-state ordering, redox energetics and reaction barriers. Coupled-cluster and large active-space treatments are the accurate fallback and do not scale to realistic active sites. These errors propagate upward, because the rate constants and binding free energies they produce are what the network-scale models consume. The furthest hardware result the paper cites is a 303-atom Trp-cage miniprotein, fragmented by wave-function embedding and solved with sample-based quantum diagonalisation on IBM Heron hardware, with a largest fragment of 33 orbitals and relative conformer energies benchmarked against CCSD.

At the network scale the problems are combinatorial or statistical

Flux balance analysis is a linear or quadratic programme solved in O(N^2.3)–O(N³) in the number of reactions. Adding regulatory logic or gene knockouts makes the problem combinatorial and NP-hard. Stochastic formulations are difficult in a different way: SSA costs scale with reaction channels times reaction events, statistical convergence needs O(1/ε²) samples, and rare-event timing scales as O(1/δ) in the spectral gap of the underlying Markov process. Each of these three has a quantum counterpart with better scaling in principle, namely quantum convex optimisation, Grover-type search, amplitude estimation reducing sampling to O(1/ε), and quantum walks improving mixing to O(1/√δ).

At the spatial scale the kernel is a large stiff linear solve

The dominant cost at every timestep is a large, stiff linear solve. Quantum linear system algorithms scale as O(log N s² κ log(1/ε)) for sparse, well-conditioned operators, which is the source of the exponential-looking speedups. Quantum systems are linear, so nonlinear PDEs have to be transformed first, using Carleman linearisation, Koopman–von Neumann formulations or level-set methods, each with its own conditions. There is no whole-cell spatial demonstration on hardware. What exists is a 5,043-dimensional linear system from an unsteady acoustic-wave PDE, solved on a superconducting processor with an error-suppressed iterative QLSA scaled up by a classical subspace method.

The advantage appears only above 10¹¹ dimensions

The Perspective works through wall-clock estimates for representative systems. Classical figures assume a 500 GFLOPS workstation or Frontier at 1.6 EFLOPS. The quantum figures assume an idealised device at 10.5 kop/s of 16-bit throughput, oracle access, sparsity s = 1, condition number κ = 1, and no data-loading cost.

Problem	Classical (Frontier)	Quantum, idealised
Core carbon metabolism FBA, N = 10⁵	0.20 µs	30.1 ms
E. coli iML1515 FBA, N = 10⁹	5.2 min	3.01 s
Yeast8 FBA, N = 10¹⁰	17.4 hrs	9.52 s
Yeast non-spatial ODEs, 15,000 species	8 µs	110 min
Yeast spatial PDEs, 6.3 × 10¹¹ coupled ODEs	~26 days	~5 hrs
E. coli RDME, one cell cycle	13 hrs – 5.5 days	~0.02 s

Two of those rows favour the classical solver, and they are the informative ones. For core carbon metabolism the problem is small enough that a classical solver on any machine finishes before a quantum device has paid its constant factors. For the non-spatial yeast ODE system, 15,000 coupled equations is not large enough for logarithmic scaling to overtake O(N^1.5), and the paper states that the same holds for a mammalian cell at an assumed 60,000 species. The advantage appears only once spatial discretisation pushes the system dimension into the 10¹¹–10¹⁶ range, which is also the regime where loading the data onto the device becomes the binding constraint.

Stochastic sampling is accounted for separately and lands the same way. A single trajectory of the E. coli glycolysis pathway is about 10¹⁴ operations, up to 42 days on one optimised workstation core, against about 14 operations per trajectory for a quantum linear solver formulation. Estimating a mean protein count in a gene regulatory circuit to 5% relative error takes about 400 classical Monte Carlo samples and about 20 with quantum Monte Carlo. Observing 100 switching events in a bistable gene circuit with a 1,000-minute switching time takes about 70 days of simulated time classically and 2.2 days with a quantum walk. All three are large systems, and none is a small system turned around.

The near-term setup is a classical solver with a QPU attached

The conclusion is not a quantum whole-cell simulator. It is a classical HPC solver with a QPU attached as an accelerator, in the way GPUs are already used in whole-cell modelling, with AI/ML surrogates alongside. Kernels are identified in advance, covering quantum chemistry, stochastic sampling, optimisation and differential equation solves. Each gets a complexity threshold, and offloading happens only above that threshold. Below it, classical computation wins once I/O and QPU latency are counted. The quantum result comes back as samples or a small number of observables rather than a full state vector, because full state tomography would consume the advantage.

Citation

Gopalakrishnan Meena, M., Kishore, D., Parks, J. M., Bertels, L., Asthagiri, D. N., Humble, T., Beck, T. L., & Doktycz, M. J. Exploring the use of quantum computing for facilitating spatially and temporally resolved models of a biological cell. Oak Ridge National Laboratory. Manuscript under review.