From Quantum Theory to a Real Quantum Computer

I’ve been working with my high school student on quantum computing, beginning with a guided journey through Scott Aaronson’s Quantum Computing Since Democritus. We’re now turning that reading into a research project: simulating a quantum spin chain on a real quantum computer and investigating how accurately we can recover its behaviour despite hardware noise.

A spin chain is a row of interacting quantum spins. For spin-½ particles, each spin has two basis states, “up” and “down,” which we can represent with a qubit. Together, these spins can form superpositions and entangled states. A standard example is the transverse-field Ising chain:

$$H=-J\sum_{i=1}^{n-1}Z_iZ_{i+1}-h\sum_{i=1}^{n}X_i.$$

Here, the Hamiltonian $H$ describes the system’s energy and governs its evolution. $J$ sets the interaction between neighbouring spins, $h$ sets an applied magnetic field, and $X_i$ and $Z_i$ represent spin components along perpendicular axes at site $i$. This gives us a compact mathematical model of interacting quantum matter. IBM’s quantum simulation introduction walks through this example.

To turn such a model into a quantum circuit, we’re learning Trotterization. If $H=A+B$, we approximate its evolution by alternating short steps generated by $A$ and $B$:

$$e^{-iHt}\approx\left(e^{-iAt/r}e^{-iBt/r}\right)^r, \qquad \hbar=1.$$

More steps generally improve the mathematical approximation, but require more gates, giving hardware errors more opportunities to accumulate.

That trade-off is central to NISQ, short for noisy intermediate-scale quantum: hardware whose useful computations are limited by noise and which operates without full fault-tolerant error correction. Preskill’s introduction to the NISQ era sets out this challenge.

Alongside Trotterization, we’re exploring three ways to manage that noise:

• Pauli twirling randomizes how a circuit is implemented while preserving its intended action. Averaging over these versions can make systematic errors behave more like random noise.
• Dynamical decoupling inserts carefully timed pulses while qubits are idle, suppressing some unwanted interactions with their surroundings.
• Readout mitigation uses calibration measurements to adjust the final statistics for errors such as the device reporting a 1 when it should report a 0.

These methods address different problems, and their effectiveness depends on the circuit and hardware. IBM’s overview explains how they work.

Our aim is to assess which approaches, alone or in combination, improve the simulation and what they cost in extra measurements or circuit operations. For small chains, classical calculations can provide a reference against which to check the quantum results.

For me, this is an exciting step in the teaching process: helping my student move from understanding quantum algorithms on paper to asking what physics we can reliably learn from an imperfect quantum device.