The NISQ Era

What Is NISQ?

NISQ (Noisy Intermediate-Scale Quantum) is a term coined by John Preskill (2018) to describe the current era of quantum computing:

• Noisy: qubits have significant error rates ( to )
• Intermediate-Scale: 50-1000+ qubits
• No fault tolerance: error correction is not yet practical

NISQ devices cannot run algorithms like Shor's, but may achieve practical advantages for specific tasks.

Current Hardware Landscape

Quantum Volume is a holistic metric that accounts for qubit count, error rates, and connectivity. Current leaders achieve QV ~ 2^{10} to 2^{12}.

Error Mitigation vs Error Correction

Error correction encodes logical qubits into many physical qubits. It requires overhead of 10^3 to 10^4 physical qubits per logical qubit.

Error mitigation reduces the impact of errors without the full overhead:

• Zero-noise extrapolation (ZNE): run at multiple noise levels, extrapolate to zero noise
• Probabilistic error cancellation (PEC): represent ideal operations as quasi-probability distributions
• Clifford data regression (CDR): learn error model from Clifford circuits
• Dynamical decoupling: apply pulse sequences to suppress decoherence

Error mitigation trades sampling overhead for reduced error.

Quantum Volume

Quantum Volume (QV) is defined as the largest random circuit of depth that can be solved with probability > 2/3 using qubits:

For QV = 2^k, the device must solve random circuits with k qubits, depth k, arbitrary connectivity, and random gates.

QV captures the tradeoff between qubit count and quality.

Random Circuit Sampling

Random circuit sampling (RCS) is the task used to demonstrate quantum supremacy:

1. Apply random single-qubit gates and fixed entangling gates in a 2D grid
2. Measure all qubits
3. Verify that the output distribution matches quantum predictions

The Google Sycamore experiment (2019) used 53 qubits and depth 20, claiming quantum supremacy. Better classical algorithms have since reduced the classical simulation cost.

NISQ Algorithms

The main NISQ algorithm candidates for near-term advantage:

1. VQE: ground state energy estimation for quantum chemistry
2. QAOA: combinatorial optimization
3. Quantum kernels: machine learning with quantum feature maps
4. Variational quantum simulation: time evolution of quantum systems

Key challenge: all these algorithms face barren plateaus and noise that may prevent scaling to useful problem sizes.

Python: Noise Simulation

import numpy as np

def depolarizing_noise(state, p):
    # Apply depolarizing noise channel to a density matrix.
    n = len(state)
    rho = np.outer(state, state.conj())
    return (1 - p) * rho + (p / n) * np.eye(n, dtype=complex)

def zne_extrapolation(values, factors):
    # Zero-noise extrapolation using Richardson extrapolation.
    coeffs = np.polyfit(factors, values, len(factors) - 1)
    return np.polyval(coeffs, 0.0)

noise_levels = [0.01, 0.02, 0.03]
exact_value = 0.5
noisy_values = [exact_value * (1 - p) for p in noise_levels]
extrapolated = zne_extrapolation(noisy_values, noise_levels)

print(f"Exact: {exact_value:.6f}")
print(f"Noisy values: {[f'{v:.6f}' for v in noisy_values]}")
print(f"ZNE extrapolated: {extrapolated:.6f}")

This demonstrates zero-noise extrapolation for error mitigation.

NISQ Algorithm Comparison

Error Mitigation Overhead

The sampling overhead for error mitigation:

The overhead must be balanced against the accuracy improvement for practical applications.