Quantum Simulation and Chemistry

In 1982 Richard Feynman observed that nature is quantum mechanical, so simulating it faithfully on classical hardware carries an exponential cost, and proposed building quantum hardware to do the job natively. The arithmetic behind that observation is short. A quantum system of n two-level parts (electron spins, for example) is described by amplitudes: complex numbers, one per configuration, whose squared magnitudes give probabilities. There are 2ⁿ configurations.

Worked example, step by step. Take n = 50 spins. The state needs 2⁵⁰ = 1,125,899,906,842,624 amplitudes, about 1.1 × 10¹⁵. Each complex amplitude stored in double precision takes 16 bytes (two 8-byte floats). Total memory: 1,125,899,906,842,624 × 16 = 18,014,398,509,481,984 bytes ≈ 18 petabytes, beyond any single machine on Earth, for just 50 spins. At n = 60 it is a thousand times more. A quantum computer represents the same state with 50 qubits (quantum bits), because the qubits simply are a quantum system of the same kind.

This is the one application where the data loading problem from chapter 1 largely disappears: the 'input' is a short description of a molecule (which atoms, where), not a billion-record database. That is why chemistry and materials pass filter question 2 while search and machine learning struggle with it.

Molecules are described by electrons occupying spin-orbitals: the discrete slots quantum mechanics allows electrons to sit in. The standard translation to qubits is the Jordan-Wigner mapping, and one honest paragraph covers what you need: assign one qubit per spin-orbital, with qubit value 1 meaning 'an electron occupies this slot' and 0 meaning empty. Electrons are antisymmetric (swapping two of them flips the sign of the state), and encoding that sign bookkeeping requires that operations on orbital k also touch chains of Z operations across qubits 0 through k−1, which lengthens circuits. The mapping is exact, not an approximation; its price is circuit length, not accuracy.

Concrete count: molecular hydrogen (H₂) in the smallest standard basis has 4 spin-orbitals, so it needs 4 qubits. The molecule's energy operator, the Hamiltonian, becomes a sum of simple terms built from Pauli operations (the X, Y, Z gates you met in X, Y, Z gates), and the number of terms grows roughly with the fourth power of the orbital count, so bigger molecules mean many more measurements, not just more qubits.

Chemistry cares about energy, above all the lowest ('ground state') energy, because energy differences decide reaction rates and binding. A quantum computer estimates energy as an expectation value: the long-run average of a measured quantity over many repeated runs, called shots.

The recipe, used by the variational quantum eigensolver (VQE): prepare a trial state with a parameterized circuit, then for each Hamiltonian term measure the qubits in the right basis, meaning the right choice of question to ask each qubit. Measuring a qubit normally asks 'are you 0 or 1?' (the Z basis). To ask the X-basis question instead, you apply an H gate just before measuring; that rotates the X question into the Z question the hardware answers, exactly as in measurement strategies. Each shot returns one bitstring; from the counts you compute averages; from the averages, energy. No single shot tells you the energy. The two runnable sections below do this for real on a 2-qubit entangled state, one whose qubits' outcomes are correlated more strongly than any classical mechanism allows (see entanglement).

Take a toy 2-qubit Hamiltonian with illustrative, made-up coefficients (units arbitrary): E = −1.0·<ZZ> − 0.5·<XX>. Suppose your two runs above returned these counts out of 1000 shots each:

• Z-basis run: 00 → 493, 11 → 507, 01 → 0, 10 → 0. Estimate: <ZZ> = (493 + 507 − 0 − 0)/1000 = +1.000 (00 and 11 contribute +1; 01 and 10 would contribute −1).
• X-basis run: 00 → 511, 11 → 489, others 0. Estimate: <XX> = (511 + 489)/1000 = +1.000.

Energy estimate: E = (−1.0)(+1.000) + (−0.5)(+1.000) = −1.50.

Second worked number: statistical precision. An average estimated from N shots has a statistical uncertainty ('shot noise') of at most about 1/√N. For N = 1000: 1/√1000 = 1/31.62 ≈ 0.032. Chemistry often needs energies to about 0.0016 hartree ('chemical accuracy'; the hartree is the standard unit of atomic-scale energy). Getting 1/√N down to 0.0016 already needs N around 400,000, and the Hamiltonian's coefficients multiply the noise on top of that, a relative precision that pushes N into the millions per term, multiplied by thousands of terms for real molecules. Shot count, not qubit count, is a leading cost in quantum chemistry, an oracle-cost issue exactly in the sense of chapter 1's filter.

Applying the reality classifications this site uses everywhere:

• Proved theory: quantum computers can simulate quantum systems with resources growing polynomially, not exponentially, in system size.
• Hardware demonstrated: ground-state energies of small molecules (H₂-scale, a handful of qubits) computed on real devices since 2014, with accuracy relying heavily on error mitigation, classical post-processing that estimates the effect of noise and subtracts it from the results rather than fixing errors during the run (a weaker tool than error correction). These molecules are trivially solvable classically; the demos validate the method, not an advantage.
• Not demonstrated: any chemistry result on a quantum computer that beats the best classical methods. Classical quantum-chemistry codes are excellent and improving (filter question 5), and they set a high bar.
• Roadmap claim: drug discovery and materials breakthroughs from quantum simulation. Plausible direction, unproven timeline; treat vendor dates as marketing until published benchmarks exist.

Our drug discovery and protein folding pages track specific claims with sources. Note that protein folding is mostly a classical optimization problem, not a quantum simulation problem, a distinction hype routinely blurs.

Running the circuits above on a real device adds noise: the exact zeros on 01 and 10 become small nonzero counts, and expectation values shrink toward 0, which biases energies. How much depends on two-qubit gate error and on how long qubits hold their state, the coherence times T1 and T2 (explained here). Deep chemistry circuits are exactly the kind that two-qubit fidelity limits: at an archetype 0.5% error per entangling gate, a 200-entangling-gate trial state already fails more often than it succeeds (0.995²⁰⁰ ≈ 0.37). Compare devices on the metrics that matter for VQE-style workloads at hardware compare, and check sourced per-device figures on the QPU pages. All of them carry methodology labels, because a vendor-claimed fidelity and an independently benchmarked one are different facts.