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Single qubit

This tutorial introduces the core concepts of quantum metrology using squint. Let's build a simple quantum phase estimation protocol step-by-step for a single qubit.

In quantum phase estimation, we want to estimate an unknown parameter \(\varphi\) that appears as a phase rotation in our quantum system. The protocol consists of four stages:

  1. Prepare a metrologically-useful probe state
  2. Interact the probe with the unknown parameter \(\varphi\)
  3. Measure the perturbed probe state
  4. Estimate \(\varphi\) from the measurement data

The initial state is,

\[|\psi\rangle = |0\rangle \]

The quantum state evolves as,

\[|\psi(\varphi)\rangle = H \cdot R_z(\varphi) \cdot H \cdot |0\rangle\]

After the transformations, the state becomes,

\[|\psi(\varphi)\rangle = \cos(\varphi/2)|0\rangle + i\sin(\varphi/2)|1\rangle\]

Measuring in the \(X\) basis (i.e., applying an H gate and measuring in the computational basis), the measurement probabilities are,

\[p(0|\varphi) = \cos^2(\varphi/2), \quad p(1|\varphi) = \sin^2(\varphi/2)\]

Build a sensor in squint

import equinox as eqx
import jax
import jax.numpy as jnp
import matplotlib.pyplot as plt
from squint.circuit import Circuit
from squint.ops.dv import DiscreteVariableState, HGate, RZGate
from squint.utils import print_nonzero_entries, partition_op

# Create a simple one-qubit phase estimation circuit
# |0⟩ --- H --- Rz(φ) --- H --- |⟩
circuit = Circuit(backend="pure")
circuit.add(DiscreteVariableState(wires=(0,), n=(0,)))          # |0⟩ state
circuit.add(HGate(wires=(0,)))                                  # Hadamard gate
circuit.add(RZGate(wires=(0,), phi=0.0 * jnp.pi), "phase")      # Phase rotation
circuit.add(HGate(wires=(0,)))                                  # Second Hadamard

# Compile the circuit for simulation
dim = 2  # qubit dimension
params, static = partition_op(circuit, "phase")
sim = circuit.compile(static, dim, params, optimize="greedy").jit()

Calculating the Fisher Information

The Fisher Information quantifies how much information our measurements contain about \(\varphi\):

# Calculate quantum and classical Fisher Information
qfi = sim.amplitudes.qfim(params)       # Quantum Fisher Information
cfi = sim.probabilities.cfim(params)    # Classical Fisher Information

print(f"Quantum Fisher Information: {qfi}")
print(f"Classical Fisher Information: {cfi}")

The Quantum Fisher Information (QFI) for a pure state \(|\psi(\varphi)\rangle\) is:

\[\mathcal{I}_\varphi^{(Q)} = 4(\langle\partial_\varphi\psi|\partial_\varphi\psi\rangle - |\langle\psi|\partial_\varphi\psi\rangle|^2)\]

The Classical Fisher Information (CFI) for measurement probabilities \(p(s_i|\varphi)\) is:

\[\mathcal{I}_\varphi^{(C)} = \sum_i \frac{(\partial_\varphi p(s_i|\varphi))^2}{p(s_i|\varphi)}\]

Visualizing the results

# Sweep through different phase values
phis = jnp.linspace(-jnp.pi, jnp.pi, 100)
params = eqx.tree_at(lambda pytree: pytree.ops["phase"].phi, params, phis)

probs = jax.vmap(sim.probabilities.forward)(params)
qfims = jax.vmap(sim.amplitudes.qfim)(params)
cfims = jax.vmap(sim.probabilities.cfim)(params)

# plot results
colors = sns.color_palette("Set2", n_colors=jnp.prod(jnp.array(probs.shape[1:])))
fig, axs = uplt.subplots(nrows=2, figsize=(6, 4), sharey=False)

for i, idx in enumerate(
    itertools.product(*[list(range(ell)) for ell in probs.shape[1:]])
):
    axs[0].plot(phis, probs[:, *idx], label=f"{idx}", color=colors[i])
axs[0].legend()
axs[0].set(xlabel=r"Phase, $\varphi$", ylabel=r"Probability, $p(\mathbf{x} | \varphi)$")

axs[1].plot(phis, qfims.squeeze(), color=colors[0], label=r"$\mathcal{I}_\varphi^Q$")
axs[1].plot(phis, cfims.squeeze(), color=colors[-1], label=r"$\mathcal{I}_\varphi^C$")
axs[1].set(
    xlabel=r"Phase, $\varphi$",
    ylabel=r"Fisher Information, $\mathcal{I}_\varphi$",
    ylim=[0, 1.05 * jnp.max(qfims)],
)

Key takeaways

  • The Fisher Information tells us how precisely we can estimate \(\varphi\)
  • Higher Fisher Information means better estimation precision
  • The Cramér-Rao bound gives us: \(\Delta^2\bar{\varphi} \geq 1/\mathcal{I}_\varphi\)