A perspective on the current state-of-the-art of quantum computing for drug discovery applications

1 Introduction

• Exact or nearly exact solutions for chemically relevant systems become intractable beyond approximately 30 electrons, motivating methods that can deliver exact quantum-mechanical solutions at manageable cost.The introduction identifies quantum computers as suited to simulating quantum-mechanical problems without exponential memory growth.Source: §1 · p. 2
• Quantum error-correction experiments have suppressed errors while preserving a logical qubit and enabling simple logical gates across superconducting, trapped-ion, and nuclear-spin platforms.The perspective describes this progress as groundwork for large-scale fault-tolerant quantum computation.Source: §1 · p. 3
• The perspective traces a pharmaceutical quantum-computing workflow from mapping the chemical problem and choosing an algorithm to specifying an error-corrected architecture, illustrated with covalent Ibrutinib bound to Bruton’s tyrosine kinase.It estimates resources for fully quantum simulations of progressively larger clusters of the binding pocket and inhibitor.Source: §1 · p. 3
• For active spaces of around 50 orbitals, recent algorithmic advances enable fully quantum calculations in a few days on sufficiently large error-corrected quantum computers.The perspective estimates resources for progressively larger clusters of the Ibrutinib binding pocket and inhibitor.Source: §1 · p. 3
• Quantum phase estimation scales more favorably than the variational quantum eigensolver for finding electronic ground-state energies, so the paper focuses on phase estimation.The perspective compares Trotterisation and qubitisation as methods for constructing the unitary operator used by phase estimation.Source: §1 · pp. 3–4

2.1 Chemistry and the electronic structure problem

• First-principles equations are insoluble beyond the simplest cases; molecular electronic states and energies are represented as eigensolutions of a Hamiltonian.The Born–Oppenheimer approximation separates electronic and nuclear degrees of freedom.Source: §2.1 · p. 5
• Hartree–Fock can handle molecules with several hundred atoms but omits electron correlation; polynomial-scaling CCSD is successful when correlation is weak.Source: §2.1 · p. 6
• FCI scales exponentially with electron and orbital counts, and classical methods remain limited by the active-space sizes needed for strongly correlated systems.CASCI restricts FCI to a complete active space, but this can still leave important problems beyond classical reach.Source: §2.1 · pp. 6–7
• The perspective aims to estimate quantum resources for a protein–drug system in which weak hydrogen-bond interactions play an important role.Source: §2.1 · p. 7

2.2 Quantum computation

• Quantum phase estimation was introduced by Kitaev in 1995 to study the Abelian stabiliser problem, and the passage identifies it as key to studying chemistry efficiently on quantum computers.The text places this development in the late 1990s.Source: §2.2 · p. 8
• QPE can estimate chemical-Hamiltonian eigenvalues to a specified accuracy with runtime polynomial in system size, motivating calculations beyond classical reach.Source: §2.2 · p. 8

2.3 The qubit Hamiltonian

• Chemical active-space Hamiltonians are expressed in second quantization, then mapped, for example by Jordan–Wigner, to Pauli-string sums for quantum algorithms.Source: §2.3 · p. 10

2.4 Algorithms

• The two main quantum algorithms for computational chemistry are VQE and QPE, and this paper focuses on QPE resource estimates for pharmaceutical calculations.Source: §2.3 · p. 11, §2.4 · p. 11

2.4.1 The variational quantum eigensolver

• VQE [12] combines a classical optimizer with a parameterized ansatz circuit, repeatedly measuring Hamiltonian expectation values and updating parameters until convergence.Source: §2.4.1 · pp. 11–12

2.4.2 Quantum phase estimation

• QPE [48,62] uses deeper circuits but typically only a handful of runs, needs no ansatz, and estimates eigenvalues directly, with precision set by data-register size.The state register begins in a state intended to overlap significantly with the ground state; Hartree–Fock is a common choice before controlled unitaries are applied.Source: §2.4.2 · pp. 13–14
• After controlled operations, inverse quantum Fourier transform and measurement of the data register yield an eigenphase estimate, whose probability is the initial state’s overlap probability with that eigenstate.Source: §2.4.2 · p. 14

3.1 Algorithm scaling

• The scaling estimates motivate algorithm choices by expressing resource requirements through key parameters while ignoring coefficients.The section frames time in terms of circuit depth, repetitions, and available parallel quantum processors, and defines chemical-system parameters including spin orbitals and electrons.Source: §3.1 · p. 14

3.1.1 VQE resources

• The VQE resource estimate assumes one qubit per spin orbital and a fixed UCCSD ansatz, which may fail to prepare a sufficiently accurate ground state for strongly correlated systems.The section notes that UCCSD uses single and double excitations from the Hartree–Fock state.Source: §3.1.1 · pp. 16–17
• VQE estimates use a favorable assumption for optimization evaluations and count separate Pauli measurements, although commuting-group measurement and integral factorization can reduce measurement requirements.The section says QPE-inspired methods can improve this scaling by increasing circuit depth, which is unlikely to be feasible in the pre-error-correction NISQ era.Source: §3.1.1 · pp. 18–19

3.1.2 QPE resources

• QPE resource requirements can be estimated for a given chemical system, but depend on the initial-state overlap with the ground state, η.The section presents a rough scaling of the time required for QPE.Source: §3.1.2 · p. 20
• QPE uses approximately n_o qubits to store the state plus auxiliary qubits; iterative phase estimation reduces energy-estimate bits to one, while recent unitary methods require about O(n_o) additional qubits.Source: §3.1.2 · p. 20
• QPE’s deep circuits are expected to require error correction; under the surface code, spatial overhead scales as d^2 and temporal overhead as d.The overheads are defined as θS and θT, respectively.Source: §3.1.2 · p. 21

3.1.3 Comparison and Discussion

• QPE is expected to become preferable to VQE once systems are large enough, but the crossover size depends on scaling constants omitted from the analysis.The comparison motivates the authors’ choice of QPE for the remainder of the work.Source: §3.1.3 · p. 22
• The scaling comparison is not definitive: VQE may improve, but its ansatz may require many parameters and circuit depth, and its optimizer may converge to a local minimum.Source: §3.1.3 · pp. 22–23

3.2 QPE in this work

• The work estimates QPE resources from the chemical system and desired accuracy as inputs, while allowing algorithmic choices.Source: §3.2 · p. 23
• A single QPE run may miss the target accuracy by returning another eigenstate, finite-bit error, or a logical error, so runs must be repeated to meet the desired success probability.The paper does not explicitly calculate the required number of repetitions, but outlines one possible method in Appendix A.Source: §3.2 · pp. 23–24

4.1 Quantum error correction and the surface code

• Useful circuits may contain over 10^10 logical gates, making error rates low enough to run them without quantum error correction unrealistic.The section concludes that quantum error correction is not optional for large-scale quantum computing applications.Source: §4.1 · p. 24
• A surface-code logical qubit uses a d × d data grid plus d^2 syndrome qubits, totaling 2d^2 physical qubits.Stabiliser measurements check and correct errors without destroying information encoded in the logical qubit.Source: §4.1 · p. 25
• The surface code has a 1% physical-error threshold; below it, increasing code distance d decreases logical error, estimated as approximately 0.1(100p)(d+1)/2 per operation.Here, p is the probability of an error per physical-qubit operation.Source: §4.1 · p. 25

4.2 Magic state factories and the QPU architecture

• The 15-to-1 factory uses 11 logical qubits and 11 time steps to produce one magic state; the 20-to-4 uses 14 logical qubits and 17 time steps to produce four.A time step is one error-corrected logical operation.Source: §4.2 · p. 26
• Concatenating eleven 15-to-1 factories with another 15-to-1 stage produces a 225-to-1 factory in 15 time steps, with failure probability 1500625p9.The process uses 225 imperfect magic states to produce one higher-quality state.Source: §4.2 · p. 26
• In the fast block, one magic state can be consumed by any data qubits within one time step, defined as d code cycles.The layout arranges data qubits in a 2D grid with auxiliary qubits between each row.Source: §4.2 · p. 26

4.3 Error-corrected resource estimation

• To keep total distillation failure probability at or below Pdf, select a factory with failure rate q ≤ Pdf/NT, where NT is the number of T gates.The required number of distilled states equals the number of T gates in the algorithm.Source: §4.3 · pp. 27–28
• One magic state is supplied per time step on average: the 15-to-1, 116-to-12, and 225-to-1 factories require 11, 5, and 15 factories, respectively.Source: §4.3 · p. 28
• Because factory placement is NP-hard [94], a greedy algorithm places connected factories sequentially, minimizing arrangement perimeter and encouraging a roughly rectangular, tightly packed layout.Source: §4.3 · pp. 28–29
• For an overall target failure rate Ptarget, solving the equation gives the required surface-code distance d; resource estimates set distillation failure probability Pdf = 1 × 10−3 and surface-code failure probability Ptarget = 9 × 10−3.Source: §4.3 · pp. 29, 31
• Each logical qubit comprises 2d^2 physical qubits, and runtime is estimated as NT × d × t, where t is the time per code cycle.Source: §4.3 · pp. 29, 31

5.1 Trotterisation

• Trotterisation divides evolution time into steps of duration τ; its approximation error vanishes as τ→0, but implementation cost rises as τ shrinks.The method therefore seeks the largest step size that meets the target accuracy.Source: §5.1 · pp. 31–32
• For the small-molecule dataset, the ground-state energy difference ϵ0 was modelled quadratically in τ, with fitted parameters a = 1.51 ± 0.84 and b = −4.66 ± 0.27.The dataset comprised H2, H+BeH2, and H2O in the STO-3G basis; the fit excluded small-τ values that deviated from quadratic behaviour, attributed to numerical error.Source: §5.1 · pp. 32–33
• For chemically accurate Trotterisation, the T-gate count is around 4 × 10^14, versus around 10^10 for linear-combination-of-unitaries methods, with projected runtimes on the order of a few days.The Trotter T-gate counts result in impractically long runtimes.Source: §5.1 · pp. 33–34

5.2 Qubitisation

• Qubitisation avoids implementing time evolution directly: QPE on a walk operator retrieves Hamiltonian energies, and the operator is built from an LCU using PREPARE/SELECT.The walk operator can require many fewer T gates than Trotterised evolution for a given Hamiltonian and target accuracy.Source: §5.2 · p. 34
• The qubitisation estimate uses a total error budget of 1.6 mHa, apportioned as 0.3 mHa truncation, 0.3 mHa PREPARE, and 1 mHa QPE error.The authors consider a sparse method tailored to Jordan–Wigner encoding.Source: §5.2 · p. 35
• The sparse qubitisation method truncates small Hamiltonian terms; CCSD(T)-based truncation can remove up to ∼90% of terms, lowering implementation cost.The truncation is selected so the energy difference between the original and truncated Hamiltonians stays within the truncation error budget.Source: §5.2 · pp. 35–36
• The LCU approach has a smaller PREP error and directly approximates the Hamiltonian, avoiding approximation in the exponential expansion.The passages contrast this with Trotterisation, which targets an approximate implementation of the time-evolution operator.Source: §5.2 · p. 36
• Finite-accuracy phase estimation contributes an error that determines how many times the walk operator must be repeated.The stated repetition count is ⌈πEmax/(2ϵQPE)⌉.Source: §5.2 · p. 36
• Qubitisation algorithms have a tremendous runtime advantage over Trotterisation.Sparse-qubitisation resource estimates are presented as directly comparable to Trotterisation runtime requirements.Source: §5.2 · p. 37

6.1 Computational Chemistry in Pharma

• Traditional force fields omit polarization, charge transfer, aromatic stacking, and metal-ion effects, and cannot describe covalent bond breaking, transition states, or spectroscopic predictions.Source: §6.1 · p. 38
• Hybrid QM/MM treats selected binding-pocket residues and the drug quantum mechanically while simulating the remainder classically; QM-cluster isolates the drug and nearest amino acids.Source: §6.1 · pp. 38–39
• Both approaches support studies of binding and electronic processes but are generally limited to a few hundred atoms, insufficient for allosteric or other large-scale mechanisms.Source: §6.1 · p. 39

6.2 Active Space Selection

• The proposed active-space selection uses local fragment occupied orbitals and perturbation-theory natural orbitals; a secondary subtractive embedding for outside-space correlation is unnecessary for resource estimation.Source: §6.2 · p. 40

6.3 The Model System

• Ibrutinib was chosen as the model drug; it inhibits BTK and covalently binds Cys481 through Michael addition.Source: §6.3 · p. 40
• Covalent binding requires favorable positioning of reactive drug and protein groups before an electronic rearrangement forms the bond, a process most common drug-design methods cannot describe.Source: §6.3 · p. 40
• The selected cluster contains 129 atoms, including Leu408, Gly409, Thr410, Gly480, Cys481, Asn484, and four waters, and is fixed at position 3 of the piperidine ring.Source: §6.3 · p. 41, Figure 5 · p. 41

6.4 Computational Methods

• Calculations used the 1.08 Å crystal structure of covalently bound ibrutinib/BTK (PDB ID: 5P9J); missing residues were filled, hydrogens added and refined, and terminal amino-acid Cα atoms plus one ligand atom constrained.Source: §6.4 · p. 42
• Active-space integrals were calculated in ORCA with def2-TZVP; Pipek–Mezey-localized occupied orbitals were assigned to fragments using intrinsic atomic orbitals and a fragment-charge threshold above 0.95.The number of active virtual orbitals was set equal to the number of selected active occupied orbitals, using perturbation theory.Source: §6.4 · p. 42

Sparse-qubitisation runtime

• For sparse-qubitisation QPE, runtime scales approximately as n_o^4.6 across active spaces from (14e,14o) to (100e,100o), assuming 1 µs per time step and considering p = 10^-4 and 10^-3.Source: Figure 6 · p. 44

Hamiltonian truncation

• With truncation error targeted at 0.3 mHa or less, CCSD(T) truncates more Hamiltonian terms than the L2-norm criterion and yields 1.2-to-2.0-fold fewer T gates.The estimated T-gate counts for both truncation approaches are presented in Table 1.Source: §7 · p. 44, Table 1 · p. 45

Physical-qubit resources

• Although qubitisation requires more data qubits, Trotterisation uses more total physical qubits because its higher T-gate count necessitates larger code distance and magic-state factories.The (32e,32o) layouts illustrate the larger data block for qubitisation and larger magic-state factories for Trotterisation.Source: §7 · p. 45, Figure 8 · p. 47

Runtime comparison

• At p = 10^-4, sparse qubitisation takes 1.9 days for (32e,32o), compared with roughly 250 years for Trotterisation, whose runtime scales more steeply with active-space size.The comparison uses sparse qubitisation with Hamiltonian truncation and textbook Trotterisation without truncation.Source: §7 · pp. 44–45, Figure 7 · p. 46

Trotterisation example

• For the Trotterisation QPE approach, the calculation uses 82 data qubits and 3.96 × 10^14 T gates, estimating 3.6 × 10^6 physical qubits.The stated magic-state distillation ratio is 225-to-1.Source: §7 · p. 47

Code distance

• For (32e,32o) qubitisation, increasing p from 10^-4 to 10^-3 raises code distance from 15 to 32; at fixed p, it rises from 15 to 20 between the two QPE approaches.The latter change accompanies a T-gate increase from 1.1 × 10^10 to 4.0 × 10^14.Source: §7 · p. 48

8 Conclusions

• (42e,42o): for Ibrutinib, sparse qubitisation reduces the estimated ground-state runtime from over 1000 years with Trotterisation to around 7.6 days, assuming a 0.01% physical error rate and 1 µs code cycles.The comparison uses the full Hamiltonian for Trotterisation and a truncated Hamiltonian for qubitisation.Source: §8 · p. 49
• Over a year is estimated for the (100e,100o) ground-state calculation, underscoring the need for further algorithmic and quantum-error-correction improvements.The paper notes that earlier algorithmic improvements have already reduced estimated resources by orders of magnitude.Source: §8 · pp. 49, 51
• Potential runtime reductions include parallel T-gate execution, further Hamiltonian truncation such as tensor hypercontraction, and advances in quantum error correction.The resource estimates assume serial T-gate execution.Source: §8 · p. 49
• Current quantum computers remain far from non-trivial QPE calculations, with error rates still needing reduction for practical scaling.The paper notes that reported error rates are close to the code threshold.Source: §8 · pp. 49–50
• Fault-tolerant QPE may handle large active spaces, but quantum-region accuracy must be matched by a balanced environment treatment, making embedding essential.Otherwise, environmental errors could overwhelm the quantum computer’s potential improvements.Source: §8 · p. 50
• Quantum methods must be assessed against DFT for accuracy and efficiency, with QPE’s exact solution or ansatz methods’ classical wavefunction comparisons, and without significant overhead.The paper frames reasonable cost as necessary for accuracy improvements.Source: §8 · p. 50
• The paper demonstrates quantum algorithms for realistic QM-cluster drug–protein binding studies, but applying them in drug-design workflows may require rethinking processes constrained by high-throughput demands.Even relatively inexpensive DFT is infeasible across much of the computational drug-design pipeline described.Source: §8 · pp. 50–51
• Realizing quantum computing’s potential requires progress across the stack as hardware scales; physical QEC demonstrations are one example, while resource estimates assume future hardware.The paper emphasizes that practical implementation must accompany improved resource estimates and hardware engineering.Source: §8 · p. 51

• Repeat QPE l times and take the median of the lowest k outcomes; the choices depend on the chemical system and calculation, and the median reduces outlier effects.Source: A Number of repetitions of QPE · p. 52
• The calculation uses P0, the probability of an eigenvalue estimate being within 2^-t of the desired eigenvalue, η, the initial-state overlap estimate, and Pf, the per-run error-correction failure probability.Source: A Number of repetitions of QPE · p. 52
• It assumes error-free excited-state measurements fall above the desired range, ground-state measurements fall within it with probability P0, and all other measurements fall below it.Treating all other measurements this way is deliberately likely worse than reality, to avoid underestimating the required repetitions.Source: A Number of repetitions of QPE · p. 52
• The method selects the smallest l and a corresponding k for which the median estimate has at least the specified success probability of being within the desired range.Source: A Number of repetitions of QPE · p. 52
• Using a larger true η than the estimate used to calculate k does not necessarily raise overall success probability, so l and/or k may need to increase across the desired η range.Source: A Number of repetitions of QPE · pp. 52–53

A.1 QPE probabilities

• The target is the probability that one eigenvalue estimate falls within ϵ = 2^-t of the true eigenvalue, given m measured bits of precision.The derivation assumes each eigenvalue satisfies 0 ≤ E_j < 1.Source: §A.1 · pp. 53–54
• The derivation represents each eigenvalue using binary digits and a residual term δ_j satisfying 0 ≤ δ_j < 1.It also defines probabilities p_l for obtaining the index of the desired eigenvalue, conditional on measuring that eigenvalue.Source: §A.1 · p. 53
• The probability of an estimate within 2^-t is formed by summing the probabilities of the corresponding eigenvalue-index outcomes.The section notes that this sum has a number of terms exponential in m − t, but says it is not prohibitive to evaluate for small values.Source: §A.1 · pp. 53–54
• For larger values, the section derives a bound that avoids the sum, invoking the Basel problem.The probability expression is described as a decreasing function of m, with a limit considered as m approaches infinity.Source: §A.1 · pp. 53–54