Quantum Simulation

Using quantum computers to simulate quantum systems. Potentially the most impactful near-term application.

Quantum simulation uses controllable quantum systems to simulate other quantum systems. This is widely considered the most promising application of quantum computing, with potential breakthroughs in chemistry, materials science, and physics.

Feynman’s Vision

Richard Feynman (1982) observed that simulating quantum systems on classical computers is exponentially hard:

“Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical.”

A quantum computer naturally represents quantum states, potentially offering exponential speedup for simulation.

Types of Quantum Simulation

Analog Simulation

Map the target system directly onto quantum hardware:

• Target Hamiltonian ↔ Hardware Hamiltonian
• Let system evolve naturally
• Examples: Cold atoms simulating condensed matter

Digital Simulation

Use quantum gates to simulate time evolution:

• Decompose evolution into gate sequences
• Trotter-Suzuki decomposition
• More flexible but more overhead

Hybrid Approaches

Combine analog and digital methods for best of both worlds.

Simulating Hamiltonians

Given Hamiltonian $H$, simulate evolution $U(t)=e^{-iHt}$:

Trotter Formula

$$e^{-i(A+B)t}\approx (e^{-iAt/n}{e}^{-iBt/n}{)}^n$$

Decompose $H$ into easy-to-simulate terms, apply in sequence.

Product Formulas

Higher-order decompositions for better accuracy:

• First-order: Error $O(t^2/n)$
• Second-order: Error $O(t^3/n^2)$
• Higher orders available

Modern Methods

• Quantum signal processing
• Qubitization
• Linear combination of unitaries (LCU)

Applications

Quantum Chemistry

• Molecular ground state energies
• Reaction dynamics
• Drug discovery
• Catalyst design

Materials Science

• High-temperature superconductivity
• Topological materials
• Battery chemistry

Fundamental Physics

• Lattice gauge theories (QCD)
• Quantum field theory
• Black hole physics

Biology

• Protein folding (quantum aspects)
• Photosynthesis
• Enzyme catalysis

Resource Estimates

Current State

• NISQ era: Small molecules, proof-of-concept
• Near-term goal: Beyond classical simulation capabilities
• Long-term: Industrial-scale chemistry/materials

Why It Matters

Classical simulation costs scale exponentially with system size. Quantum simulation scales polynomially. This could enable:

• Designing new drugs computationally
• Room-temperature superconductors
• Better batteries and solar cells
• Understanding fundamental physics

See also: VQE, Hamiltonian, Quantum Advantage, NISQ