Entangled Sensors¶

This tutorial shows how entanglement improves measurement precision beyond the Standard Quantum Limit.

SQL vs Heisenberg Limit¶

With \(N\) independent qubits, Fisher Information adds linearly — the Standard Quantum Limit (SQL): \(\(\mathcal{I}_\varphi^{\text{SQL}} = N\)\)

With entanglement, we can achieve the Heisenberg Limit: \(\(\mathcal{I}_\varphi^{\text{HL}} = N^2\)\)

The key is the GHZ state: \(|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}(|00\cdots0\rangle + |11\cdots1\rangle)\)

When phase \(\varphi\) is encoded on each qubit, the GHZ state evolves to: \(\(|\text{GHZ}(\varphi)\rangle = \frac{1}{\sqrt{2}}(|00\cdots0\rangle + e^{iN\varphi}|11\cdots1\rangle)\)\)

The phase accumulates as \(N\varphi\), giving \(N^2\) Fisher Information.

Building the GHZ Sensor¶

import jax.numpy as jnp
from squint.circuit import Circuit
from squint.simulator.tn import Simulator
from squint.ops.base import Wire, SharedGate
from squint.ops.dv import DiscreteVariableState, HGate, CXGate, RZGate
from squint.utils import partition_op

N = 4
wires = [Wire(dim=2, idx=i) for i in range(N)]

circuit = Circuit()

# Initialize all qubits in |0⟩
for w in wires:
    circuit.add(DiscreteVariableState(wires=(w,), n=(0,)))

# GHZ preparation: H then CNOTs
circuit.add(HGate(wires=(wires[0],)))
for i in range(1, N):
    circuit.add(CXGate(wires=(wires[0], wires[i])))

# Phase encoding on all qubits (shared parameter)
circuit.add(
    SharedGate(op=RZGate(wires=(wires[0],), phi=0.0), wires=tuple(wires[1:])),
    "phase"
)

# Measurement basis
for w in wires:
    circuit.add(HGate(wires=(w,)))

SharedGate applies the same RZGate to all qubits with a shared phase parameter. This correctly accounts for the parameter appearing on all qubits when computing derivatives.

Computing Fisher Information¶

params, static = partition_op(circuit, "phase")
sim = Simulator.compile(static, params, optimize="greedy").jit()

qfi = sim.amplitudes.qfim(params)
print(f"QFI: {qfi.squeeze():.0f}")
print(f"Heisenberg limit: {N**2}")
print(f"SQL: {N}")

For \(N=4\), you should get QFI = 16 = \(N^2\).

Verifying the Scaling¶

import matplotlib.pyplot as plt

def ghz_qfi(N):
    wires = [Wire(dim=2, idx=i) for i in range(N)]
    circuit = Circuit()

    for w in wires:
        circuit.add(DiscreteVariableState(wires=(w,), n=(0,)))

    circuit.add(HGate(wires=(wires[0],)))
    for i in range(1, N):
        circuit.add(CXGate(wires=(wires[0], wires[i])))

    circuit.add(
        SharedGate(op=RZGate(wires=(wires[0],), phi=0.0), wires=tuple(wires[1:])),
        "phase"
    )

    for w in wires:
        circuit.add(HGate(wires=(w,)))

    params, static = partition_op(circuit, "phase")
    sim = Simulator.compile(static, params, optimize="greedy").jit()
    return sim.amplitudes.qfim(params).squeeze()

N_values = [1, 2, 3, 4, 5, 6]
qfi_values = [ghz_qfi(N) for N in N_values]

plt.plot(N_values, qfi_values, 'o-', label='GHZ', markersize=10)
plt.plot(N_values, [N**2 for N in N_values], '--', label='Heisenberg ($N^2$)')
plt.plot(N_values, N_values, ':', label='SQL ($N$)')
plt.xlabel('Number of qubits $N$')
plt.ylabel('Quantum Fisher Information')
plt.legend()

Summary¶

GHZ states achieve Heisenberg scaling: \(\mathcal{I} = N^2\)
SharedGate handles parameter sharing across qubits
The quadratic improvement comes from coherent phase accumulation
This scaling is fragile to noise — see Modeling Noise

Next: Optimization shows how to find optimal probe states variationally.