CostAnalysis (Encodings)

CostAnalysis Module

Namespace: Encodings
Assembly: Encodings.dll

Unified cost metrics for encoded Hamiltonians, including qubitization resource estimation.

Provides cost functions that characterise a Hamiltonian in the Pauli basis independently of any particular simulation strategy. Metrics include the LCU 1-norm (λ), Pauli weight statistics, and term counts — all of which feed into resource estimates for Trotterization, qubitization, and measurement programs.

The qubitization resource estimate follows Low & Chuang (2019): the query complexity of quantum signal processing scales as O(λ / ε), where λ = Σ|cₖ| is the LCU 1-norm and ε is the target precision.

Types

Type	Description
HamiltonianCosts	Comprehensive cost metrics for a Hamiltonian in Pauli basis.
QubitizationCosts	Qubitization resource estimate for a Hamiltonian expressed as an LCU.
SimulationCostComparison	Simulation cost comparison at a given precision target.

Functions and values

Function or value	Description
`compareCosts encodings` Full Usage: `compareCosts encodings` Parameters: encodings : `(string * PauliRegisterSequence)[]` - Array of (name, Hamiltonian) pairs. Returns: `(string * HamiltonianCosts)[]` Array of (name, costs) pairs sorted by λ-norm ascending.	Compare cost metrics across multiple encoded Hamiltonians. encodings : `(string * PauliRegisterSequence)[]` Array of (name, Hamiltonian) pairs. Returns: `(string * HamiltonianCosts)[]` Array of (name, costs) pairs sorted by λ-norm ascending.
`compareQubitizationCosts encodings` Full Usage: `compareQubitizationCosts encodings` Parameters: encodings : `(string * PauliRegisterSequence)[]` - Array of (name, Hamiltonian) pairs. Returns: `(string * QubitizationCosts)[]` Array of (name, costs) pairs sorted by λ ascending.	Compare qubitization costs across multiple encoded Hamiltonians. encodings : `(string * PauliRegisterSequence)[]` Array of (name, Hamiltonian) pairs. Returns: `(string * QubitizationCosts)[]` Array of (name, costs) pairs sorted by λ ascending.
`hamiltonianCosts hamiltonian` Full Usage: `hamiltonianCosts hamiltonian` Parameters: hamiltonian : `PauliRegisterSequence` - The Hamiltonian as a sum of Pauli strings. Returns: `HamiltonianCosts` A HamiltonianCosts record with all metrics.	Compute comprehensive cost metrics for a Hamiltonian. hamiltonian : `PauliRegisterSequence` The Hamiltonian as a sum of Pauli strings. Returns: `HamiltonianCosts` A HamiltonianCosts record with all metrics.
`pauliWeight register` Full Usage: `pauliWeight register` Parameters: register : `PauliRegister` Returns: `int`	Compute the Pauli weight (number of non-identity positions) of a register. register : `PauliRegister` Returns: `int`
`qubitizationCosts hamiltonian` Full Usage: `qubitizationCosts hamiltonian` Parameters: hamiltonian : `PauliRegisterSequence` - The Hamiltonian as a sum of Pauli strings. Returns: `QubitizationCosts` A QubitizationCosts record.	Compute qubitization resource estimates for a Hamiltonian. hamiltonian : `PauliRegisterSequence` The Hamiltonian as a sum of Pauli strings. Returns: `QubitizationCosts` A QubitizationCosts record.
`qubitizationQueries costs time epsilon` Full Usage: `qubitizationQueries costs time epsilon` Parameters: costs : `QubitizationCosts` - Pre-computed qubitization costs. time : `float` - Evolution time t. epsilon : `float` - Target precision ε. Returns: `int` Estimated query count (rounded up).	Estimate the number of queries to the block-encoding oracle needed for time evolution exp(−iHt) to precision ε. Based on the quantum signal processing bound: the number of queries scales as O(λt/ε + log(1/ε)), where λ is the LCU 1-norm. The leading term dominates for chemically relevant precisions. costs : `QubitizationCosts` Pre-computed qubitization costs. time : `float` Evolution time t. epsilon : `float` Target precision ε. Returns: `int` Estimated query count (rounded up).
`simulationCosts hamiltonian time epsilons` Full Usage: `simulationCosts hamiltonian time epsilons` Parameters: hamiltonian : `PauliRegisterSequence` - The encoded Hamiltonian. time : `float` - Evolution time t (default 1.0). epsilons : `float[]` - Array of precision targets to evaluate. Returns: `SimulationCostComparison[]` Array of cost comparisons, one per epsilon.	Compare Trotter and qubitization costs for a given Hamiltonian at multiple precision targets. Trotter step count uses the standard first-order bound: r ≥ ⌈λ²t²/(2ε)⌉, where λ = Σ\|cₖ\| and ε is the target precision. Each step contributes the CNOT count from the CNOT staircase decomposition. Qubitization query count follows the QSP bound: ⌈λt/ε⌉. The crossover precision ε* is the point where Trotter total CNOTs equals qubitization queries — below ε*, qubitization dominates. Since Trotter CNOTs per step are encoding-dependent but λ is encoding-independent, the crossover point shifts with encoding choice. hamiltonian : `PauliRegisterSequence` The encoded Hamiltonian. time : `float` Evolution time t (default 1.0). epsilons : `float[]` Array of precision targets to evaluate. Returns: `SimulationCostComparison[]` Array of cost comparisons, one per epsilon.
`trotterQubitizationRatio hamiltonian time` Full Usage: `trotterQubitizationRatio hamiltonian time` Parameters: hamiltonian : `PauliRegisterSequence` - The encoded Hamiltonian. time : `float` - Evolution time t. Returns: `float` The cost ratio (Trotter CNOTs / qubitization queries). Values >1 mean qubitization is more efficient.	Estimate the crossover precision ε* where Trotter total CNOTs equals qubitization queries. From the cost models: Trotter CNOTs = ⌈λ²t²/(2ε)⌉ × C (C = CNOTs per step) Qubitization = ⌈λt/ε⌉ Setting equal: λ²t²C/(2ε) = λt/ε → ε* = λtC/2. Wait — that doesn't depend on ε! The ratio is: Trotter/Qubitization = λtC/2. This means for first-order Trotter, the ratio is ε-independent: qubitization wins whenever λtC/2 > 1, i.e., when C > 2/(λt). For molecular Hamiltonians with large λ and C, qubitization almost always wins — but reducing C (via better encoding) narrows the gap, and for higher-order Trotter the crossover becomes ε-dependent. hamiltonian : `PauliRegisterSequence` The encoded Hamiltonian. time : `float` Evolution time t. Returns: `float` The cost ratio (Trotter CNOTs / qubitization queries). Values >1 mean qubitization is more efficient.