org.soulspace.qclojure.application.algorithm.quantum-metrics — org.soulspace/qclojure 0.24.0

Quantum metrics and distance measures for quantum states and circuits.

This namespace provides algorithms for measuring similarity and overlap between quantum states using various quantum metrics:

• SWAP Test: Measures the inner product between two quantum states
• Quantum Fisher Information Matrix (QFIM): Characterizes parameter sensitivity
• Classical Fisher Information Matrix (CFIM): Characterizes measurement sensitivity

These metrics are fundamental for quantum state tomography, quantum metrology, variational quantum algorithms, and quantum machine learning applications.

classical-fisher-information-matrix clj

(classical-fisher-information-matrix
  ansatz-fn
  backend
  parameters
  &
  {:keys [shots shift parallel? epsilon]
   :or {shift (/ fm/PI 2) parallel? false epsilon 1.0E-10}
   :as options})

Calculates the Classical Fisher Information Matrix (CFIM) from measurement data.

The CFIM quantifies parameter sensitivity using measurement outcome probabilities instead of quantum state vectors. It is defined as:

F_ij = Σ_x [1/p(x|θ)] · [∂p(x|θ)/∂θᵢ] · [∂p(x|θ)/∂θⱼ]

where x ranges over all possible measurement outcomes and p(x|θ) is the probability of observing outcome x given parameters θ.

The CFIM is suitable for real quantum hardware (QPUs) where only measurement frequencies are available, not full state vectors. It provides a lower bound on the Quantum Fisher Information (QFIM ≥ CFIM).

Use Cases:

• Parameter optimization on real quantum hardware
• Natural gradient methods when state vector access is unavailable
• Quantum metrology with shot-limited measurements
• Variational algorithms on cloud-based QPUs (IBM, Rigetti, IonQ, etc.)

Parameters:

• ansatz-fn: Function that creates a quantum circuit from parameters. Takes parameter vector, returns circuit with measurements.
• backend: Quantum backend for circuit execution (real QPU or shot-based simulator).
• parameters: Current parameter vector (vector of numbers).
• options: Map with:
  • :shots - Number of measurements (required, e.g., 8192)
  • :shift - Parameter shift for probability derivatives (default: π/2)
  • :parallel? - Use parallel computation (default: false)
  • :epsilon - Regularization for zero probabilities (default: 1e-10)
  • Additional backend execution options

Returns: Classical Fisher Information Matrix as a vector of vectors (NxN matrix). The matrix is real-valued, symmetric, and positive semi-definite.

Example:

(defn my-ansatz [params]
  (-> (circuit/create-circuit 2 "Ansatz")
      (circuit/ry-gate 0 (nth params 0))
      (circuit/ry-gate 1 (nth params 1))
      (circuit/cnot-gate 0 1)
      (circuit/measure-all-operation)))

(def cfim (classical-fisher-information-matrix 
            my-ansatz 
            qpu-backend 
            [0.5 1.0]
            {:shots 8192}))
;=> [[1.95 0.02] [0.02 1.98]]  ; With shot noise

References:

• Stokes et al., "Quantum Natural Gradient", Quantum 4, 269 (2020)
• Cerezo et al., "Variational Quantum Algorithms", Nat. Rev. Phys. 3, 625 (2021)
• Liu et al., "Parameter Estimation on Quantum Hardware", arXiv:2107.09737

quantum-fisher-information-matrix clj

(quantum-fisher-information-matrix
  ansatz-fn
  backend
  parameters
  &
  {:keys [shift parallel?] :or {shift (/ fm/PI 2) parallel? false} :as options})

Calculates the Quantum Fisher Information Matrix (QFIM) for a parameterized quantum state.

The QFIM characterizes the distinguishability of quantum states under parameter variations. It is defined as:

F_ij = 4 · Re[⟨∂ψ/∂θᵢ|∂ψ/∂θⱼ⟩ - ⟨∂ψ/∂θᵢ|ψ⟩⟨ψ|∂ψ/∂θⱼ⟩]

The QFIM provides the optimal metric for parameter updates in quantum optimization, and is used in quantum metrology to achieve the quantum Cramér-Rao bound for parameter estimation.

This implementation uses the parameter shift rule to compute state derivatives efficiently, requiring 2N circuit evaluations where N is the number of parameters.

Use Cases:

• Quantum Natural Gradient optimization (faster convergence than standard gradients)
• Quantum Metrology (optimal parameter estimation bounds)
• Variational Quantum Algorithms (VQE, QAOA optimization)
• Quantum Machine Learning (training quantum neural networks)

Parameters:

• ansatz-fn: Function that creates a quantum circuit from parameters. Takes parameter vector, returns circuit.
• backend: Quantum backend for circuit execution (simulator with state vector access).
• parameters: Current parameter vector (vector of numbers).
• options: Optional map with:
  • :shift - Parameter shift for derivatives (default: π/2)
  • :parallel? - Use parallel computation (default: false)
  • Backend execution options: :shots, :initial-state, :result-specs

Returns: Quantum Fisher Information Matrix as a vector of vectors (NxN matrix where N = parameter count). The matrix is real-valued, symmetric, and positive semi-definite.

Example:

(defn my-ansatz [params]
  (-> (circuit/create-circuit 2 "Ansatz")
      (circuit/ry-gate 0 (nth params 0))
      (circuit/ry-gate 1 (nth params 1))
      (circuit/cnot-gate 0 1)))

(def qfim (quantum-fisher-information-matrix 
            my-ansatz 
            simulator 
            [0.5 1.0]
            {:shift (/ Math/PI 2)}))
;=> [[4.0 0.0] [0.0 4.0]]  ; For independent rotations

References:

• Nielsen & Chuang, "Quantum Computation and Quantum Information", Chapter 9
• Stokes et al., "Quantum Natural Gradient", Quantum 4, 269 (2020)
• Liu et al., "Efficient Learning of Quantum States", Nat. Phys. 17, 1349 (2021)

swap-test-circuit clj

(swap-test-circuit n-qubits)

Constructs a SWAP test circuit for measuring overlap between two quantum states.

The SWAP test is a quantum algorithm that uses an ancilla qubit and a controlled-SWAP (Fredkin) gate to measure the inner product |⟨ψ|φ⟩|² between two quantum states.

Circuit structure:

1. Ancilla qubit (qubit 0) is initialized to |0⟩ and Hadamard applied
2. State 1 (|ψ⟩) occupies qubits 1 to n
3. State 2 (|φ⟩) occupies qubits n+1 to 2n
4. Controlled-SWAP (Fredkin) gates swap corresponding qubits of states 1 and 2, controlled by the ancilla
5. Final Hadamard applied to ancilla
6. Measure ancilla qubit

The probability of measuring |0⟩ on the ancilla is: P(0) = (1 + |⟨ψ|φ⟩|²)/2 From this, we can extract: |⟨ψ|φ⟩|² = 2·P(0) - 1

Parameters:

• n-qubits: Number of qubits in each state to compare (positive integer)

Returns: A quantum circuit that performs the SWAP test. The circuit has 2n+1 qubits total:

• Qubit 0: Ancilla qubit for measurement
• Qubits 1 to n: First quantum state |ψ⟩
• Qubits n+1 to 2n: Second quantum state |φ⟩

Usage: To use this circuit, prepare the initial state as |0⟩⊗|ψ⟩⊗|φ⟩ where |ψ⟩ and |φ⟩ are the states to compare. Execute the circuit and measure the ancilla qubit.

Example: (def swap-test (swap-test-circuit 2)) ; Compare two 2-qubit states ; Prepare initial state |0⟩⊗|ψ⟩⊗|φ⟩ (5 qubits total) ; Execute circuit and measure ancilla (qubit 0) ; If P(0) ≈ 1, states are identical; if P(0) ≈ 0.5, states are orthogonal

References:

• Nielsen & Chuang, "Quantum Computation and Quantum Information", Section 9.3
• Buhrman, Cleve, Watrous, de Wolf, "Quantum Fingerprinting" (2001)