Enhancements to Microsoft.Quantum.Preparation namespace

[msoeken]

Enhancements to Microsoft.Quantum.Preparation namespace

Proposal

Deprecated API

(full signatures for replacements are listed below)

• function QuantumROM(targetError: Double, coefficients: Double[]) → PurifiedMixedState
• function QuantumROMQubitCount(targetError: Double, nCoeffs: Int) → PurifiedMixedStateRequirements
• operation PrepareQubit(basis : Pauli, qubit : Qubit) → PreparePauliEigenstate
• operation PrepareArbitraryState(coefficients : ComplexPolar[], qubits : LittleEndian) → PrepareArbitraryStateCP
• function StatePreparationComplexCoefficients(coefficients : ComplexPolar[]) → only as operation PrepareArbitraryStateCP
• function StatePreperationPositiveCoefficients(coefficients : Double[]) → only as operation PrepareArbitraryStateD

New UDTs

newtype MixedStatePreparation = (
    Requirements: MixedStatePreparationRequirements,
    Norm: Double,
    Prepare: ((LittleEndian, Qubit[], Qubit[]) => Unit is Adj + Ctl)
);

newtype MixedStatePreparationRequirements = (
    NTotalQubits: Int,
    (
        NIndexQubits: Int,
        NDataQubits: Int,
        NGarbageQubits: Int
    )
);

Note that Prepare takes as second parameter an additional Qubit register to prepare auxiliary data, e.g., the sign of a coefficient.

New or modified functions and operations

M.Q.Preparation namespace

• function PurifiedMixedState(targetError : Double, coefficients : Double[]) : MixedStatePreparation
  (new name and signature for QuantumROM)
• function PurifiedMixedStateWithData(targetError : Double, coefficients : (Double, Bool[])[]) : MixedStatePreparation
• function PurifiedMixedStateRequirements(targetError : Double, nCoefficients : Int) : MixedStatePreparationRequirements
• operation PreparePauliEigenstate(basis : Pauli, qubit : Qubit) : Unit
• operation PrepareArbitraryStateCP(coefficients : ComplexPolar[], qubits : LittleEndian) : Unit is Adj + Ctl
• operation PrepareArbitraryStateD(coefficients : Double[], qubits : LittleEndian) : Unit is Adj + Ctl
• function BlochSphereCoordinates(a0 : ComplexPolar, a1 : ComplexPolar) : (ComplexPolar, Double, Double)

Update: As per API review discussions, the term "purified mixed state" is used in the context of this proposal to mean states of the form |ψ⟩ = Σᵢ √𝑝ᵢ |𝑖⟩ |garbageᵢ⟩ specified by a vector of coefficients {𝑝ᵢ}. States of this form can be reduced to mixed states ρ ≔ 𝑝ᵢ |𝑖⟩⟨𝑖| by tracing over the "garbage" register (that is, a mixed state that is diagonal in the computational basis). Similarly, a "purified mixed state with data" refers to a state of the form Σᵢ √𝑝ᵢ |𝑖⟩ |𝑥ᵢ⟩ |garbageᵢ⟩, where each 𝑥ᵢ is a bitstring encoding additional data associated with the register |𝑖⟩.

See https://arxiv.org/pdf/1805.03662.pdf?page=15 for further discussion.

M.Q.Convert namespace

• function DoubleAsComplexPolar(input : Double) : ComplexPolar

Example use cases

Prepare purified mixed state and also output sign of coefficient

let coefficients = [3.14, -1.59, -2.65, -3.58];

// extract sign and wrap it into singleton Bool array
let data = Mapped(ConstantArray(1, _), Mapped(LessThanD(_, 0.0), coefficients));

let state = PurifiedMixedStateWithData(10e-2, Zipped(coefficients, data));
let reqs = state::Requirements;
using ((indexQubits, dataQubits, garbageQubits) = (reqs::NIndexQubits, reqs::NDataQubits, reqs::NGarbageQubits)) {
  state::Prepare(LittleEndian(indexQubits), dataQubits, garbageQubits);

  let signQubit = Head(dataQubits);
  // ...
}

Considerations

• Some changes are already implemented as part of Classical Control Migrate qsharp-compiler#322
• Previous discussion on these changes in Generic function unnexpectedly returns 'QVoid' when called from another file qsharp-runtime#76 with implementations in Bug fix for attribute resolution qsharp-compiler#212 (we can reuse parts of these implementations)

[guenp]

@msoeken thanks for the proposal! I think it looks good, just some comments/questions:

• Why deprecate PrepareQubit? I think it can still be a useful operation to keep. Edit: Just noticed it's to be renamed to PrepareSingleQubitPositivePauliEigenstate, if that's the case would you mind noting it in the proposal for clarity?
• I'm in favor of branching out PrepareArbitraryState into PrepareArbitraryStateCP and PrepareArbitraryStateD, however if/when we get discriminated unions (Discriminated unions, Maybe? qsharp-compiler#406) would we need to refactor these again?
• What is the difference between PurifiedMixedState and PurifiedMixedStateAndSign?
• Why does BlochSphereCoordinates take two ComplexPolar values? Does it calculate the difference between the two on the Bloch sphere?

[cgranade]

• I'm in favor of branching out PrepareArbitraryState into PrepareArbitraryStateCP and PrepareArbitraryStateD, however if/when we get discriminated unions (microsoft/qsharp-compiler#406) would we need to refactor these again?

Personally, I'd suggest that bounded polymorphism (microsoft/qsharp-language#149) is likely a better approach than DUs for eliminating type suffixes in general across Q# APIs, such that I'd expect similar to apply here as well.

[guenp]

@cgranade Ah, yes that would also work, I personally think that syntax looks very complicated, something like coefficients: Union[ComplexPolar[], Double[]) like in Python seems easier to read/understand. But either way it looks like there will likely be a compiler feature that would simplify operations with type suffixes in the future.

[cgranade]

@cgranade Ah, yes that would also work, I personally think that syntax looks very complicated, something like coefficients: Union[ComplexPolar[], Double[]) like in Python seems easier to read/understand. But either way it looks like there will likely be a compiler feature that would simplify operations with type suffixes in the future.

As per action items in most recent API review (#368), I've opened up a suggestion for "anonymous" DUs at microsoft/qsharp-compiler#406. That language suggestion would allow for writing Q# types like ComplexPolar[] | Double[]; while such types could not be used without deconstruction into cases, they would still provide a lighter-weight solution than named DU cases (e.g.: newtype Coefficients = Complex : ComplexPolar[] | Real : Double[];), allowing for removing type suffixes in cases such as PrepareArbitraryStateCP and PrepareArbitraryStateD without requiring all of the concepts / typeclasses / bounded polymorphism suggestion at microsoft/qsharp-language#149.

[cgranade]

As a brief update, it looks like @crazy4pi314 has incorporated the anonymous DU suggestion at microsoft/qsharp-compiler#406 into a new language suggestion at microsoft/qsharp-language#51.