Quantum ComputingVerificationUniversity
AQAAPOntarioNSWCBSEGCE O-LevelMoECAPS

Quantum State Fidelity Calculator

Closeness of two quantum states.

Use the free calculatorCheck the variablesOpen the advanced solver
Open Advanced Calculator
This is the free calculator preview. Advanced walkthroughs stay in the app.
Result
Ready
Fidelity

Formula first

Overview

Quantum state fidelity is a metric used to determine the mathematical similarity between two quantum states, represented by their density matrices ρ and σ. It quantifies how indistinguishable the states are from one another, ranging from 0 for perfectly orthogonal states to 1 for identical states.

Symbols

Variables

F = Fidelity

Fidelity

Apply it well

When To Use

When to use: Use this formula when evaluating the accuracy of quantum state preparation or assessing the impact of decoherence in a quantum channel. It is the standard calculation for comparing a noisy experimental density matrix against a theoretical ideal state during quantum process tomography.

Why it matters: Fidelity serves as the primary benchmark for quantum hardware performance, providing a single scalar value that indicates how reliably a quantum computer preserves information. High fidelity is a prerequisite for the implementation of quantum error-correction and the realization of fault-tolerant quantum computation.

One free problem

Practice Problem

A quantum bit is intended to be prepared in the pure state |0⟩, but environmental noise results in a mixed state σ = 0.85|0⟩⟨0| + 0.15|1⟩⟨1|. Calculate the quantum state fidelity f between the ideal state and the actual state.

Fidelity0.85

Solve for:

Hint: When one state is pure, the fidelity is simply the expectation value of the other state's density matrix with respect to that pure state.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac L. Chuang
  2. The Theory of Quantum Information by John Preskill (Lecture Notes, Caltech)
  3. Wikipedia: Fidelity of quantum states
  4. IUPAC Gold Book: 'density matrix (density operator)'
  5. Nielsen, Michael A., and Isaac L. Chuang. 'Quantum Computation and Quantum Information.' Cambridge University Press, 2010.
  6. Wikipedia: 'Density matrix'
  7. Nielsen and Chuang Quantum Computation and Quantum Information
  8. Jozsa Fidelity for Mixed Quantum States