Partial trace

Functions for calculating reduced density matrices, creating a new output Qureg. More...

Functions
Qureg	calcPartialTrace (Qureg qureg, int *traceOutQubits, int numTraceQubits)
Qureg	calcReducedDensityMatrix (Qureg qureg, int *retainQubits, int numRetainQubits)

Detailed Description

Functions for calculating reduced density matrices, creating a new output Qureg.

Function Documentation

calcPartialTrace()

Qureg calcPartialTrace	(	Qureg	qureg,
		int *	traceOutQubits,
		int	numTraceQubits )

Creates and populates a new Qureg which is a reduced density matrix resulting from tracing out the specified qubits of qureg. This should be later freed by the user like all Qureg.

Note that the deployments of the output Qureg (i.e. whether multithreaded, GPU-accelerated and distributed) will match those of qureg. It is ergo intended that this function is used to trace out few qubits, and may show worsening performance when tracing many qubits.

The ordering of traceOutQubits has no effect, and the ordering of the remaining qubits in the output Qureg match their original relative ordering in qureg.

Formulae

Let \(\dmrho_{\text{in}} = \) qureg and let \(\vec{t} = \) traceOutQubits which is a list of length \(n = \) numTraceQubits.

This function returns a new Qureg \(\dmrho_{\text{out}}\) which satisfies

\[ \dmrho_{\text{out}} = \text{Tr}_{\vec{t}} \left( \dmrho_{\text{in}} \right) = \sum\limits_i^{2^n} (\hat{\id} \otimes \bra{i}_{\vec{t}} ) \, \dmrho_{\text{in}} \, (\hat{\id} \otimes \ket{i}_{\vec{t}} ) \]

where \(\ket{i}_{\vec{t}}\) notates the \(i\)-th basis state (in any orthonormal basis) of the targeted qubits, and \((\hat{\id} \otimes \ket{i}_{\vec{t}})\) notates interleaved identity operators upon the non-targeted qubits.

Given an \(N\)-qubit Qureg \(\dmrho_{\text{in}}\), the output \(\dmrho_{\text{out}}\) contains \(N-n\) qubits.

Constraints

• The given qureg must be a density matrix. It is however straightforward to prepare a density matrix from a statevector.

  // let qureg be the intended initial statevector
   
  Qureg temp = createDensityQureg(qureg.numQubits);
  initPureState(temp, qureg);
   
  Qureg reduced = calcPartialTrace(temp, traceOutQubits, numTraceQubits);
  destroyQureg(temp);

• When qureg is distributed, the returned Qureg will also be distributed, which imposes a minimum on the number of qubits contained within; \(\log_2(W)\) where \(W\) is the number of distributed nodes (or "world size"). This imposes a maximum upon traceOutQubits of

  numTraceQubits <= qureg.numQubits - qureg.logNumNodes

Equivalences

• The function calcReducedDensityMatrix() is entirely equivalent, but conveniently permits specifying a list of which qubits to retain during partial tracing.
• The functions setQuregToPartialTrace() and setQuregToReducedDensityMatrix() are also equivalent but permit overwriting an existing Qureg.

Example

Qureg state = createDensityQureg(5);
initRandomMixedState(state, 10);
reportQureg(state);
 
int qubits[] = {0,2,4};
Qureg reduced = calcPartialTrace(state, qubits, 3);
reportQureg(reduced);
 
// state's qubits {1,3} have become reduced's qubits {0,1}

Parameters

[in]	qureg	a density matrix which is not modified.
[in]	traceOutQubits	a list of qubits to trace out and ergo from the output Qureg.
[in]	numTraceQubits	the length of traceOutQubits.

Returns

A new, smaller Qureg initialised to the reduced density matrix of qureg.

Exceptions

error	if qureg is uninitialised. if numTraceQubits is less than one. if numTraceQubits is equal or greater than the number of qubits in qureg. if qureg is distributed and numTraceQubits exceeds qureg.numQubits - qureg.logNumNodes. if the system contains insufficient RAM (or VRAM) to store the new Qureg in any deployment. if any memory allocation of the output Qureg unexpectedly fails.
seg-fault	if traceOutQubits is not a list of length numTraceQubits.

Attention

This function's input validation has not yet been unit tested, so erroneous usage may produce unexpected output. Please use with caution!

See also

• calcReducedDensityMatrix()
• setQuregToPartialTrace()
• setQuregToReducedDensityMatrix()

Author

Tyson Jones

Definition at line 374 of file calculations.cpp.

                                                                             {
    validate_quregFields(qureg, __func__);
    validate_quregIsDensityMatrix(qureg, __func__);
    validate_targets(qureg, traceOutQubits, numTraceQubits, __func__);
    validate_quregCanBeReduced(qureg, numTraceQubits, __func__);
 
    // attempt to create reduced Qureg with same deployments as input Qureg
    Qureg out = validateAndCreateCustomQureg(
        qureg.numQubits - numTraceQubits, 
        qureg.isDensityMatrix,  qureg.isDistributed, 
        qureg.isGpuAccelerated, qureg.isMultithreaded, __func__);
 
    // set it to reduced density matrix
    auto targets = lists_getList64(traceOutQubits, numTraceQubits);
    localiser_densmatr_partialTrace(qureg, out, targets);
 
    return out;
}

Referenced by calcPartialTrace(), calcReducedDensityMatrix(), and TEST_CASE().

calcReducedDensityMatrix()

Qureg calcReducedDensityMatrix	(	Qureg	qureg,
		int *	retainQubits,
		int	numRetainQubits )

Creates and populates a new Qureg which is a reduced density matrix of qureg, retaining only the specified qubits and tracing out all others.

Note that the deployments of the output Qureg (i.e. whether multithreaded, GPU-accelerated and distributed) will match those of qureg. It is ergo intended that this function is used to preserve most qubits of qureg, and may show worsening performance when retaining only few.

Attention

The ordering of retainQubits has no effect on the output state. The ordering of the retained qubits will match their original, relative ordering in qureg.

Formulae

This function is entirely equivalent to calcPartialTrace() except that here the retained qubits are specified, whereas calcPartialTrace() accepts those to be traced out.

Let \(\dmrho_{\text{in}} = \) qureg, \(\vec{r} = \) retainQubits, and let \(\vec{q}\) be a list containing all qubits of qureg. This function partially traces out all qubits in list \(\vec{t} = \vec{q} \setminus \vec{r}\), and returns a new Qureg \(\dmrho_{\text{out}}\) which satisfies

\[ \dmrho_{\text{out}} = \text{Tr}_{\vec{t}} \left( \dmrho_{\text{in}} \right) = \sum\limits_i^{2^n} (\hat{\id} \otimes \bra{i}_{\vec{t}} ) \, \dmrho_{\text{in}} \, (\hat{\id} \otimes \ket{i}_{\vec{t}} ) \]

where \(\ket{i}_{\vec{t}}\) notates the \(i\)-th basis state (in any orthonormal basis) of the qubits in \(\vec{t}\), and \((\hat{\id} \otimes \ket{i}_{\vec{t}})\) notates interleaved identity operators upon the qubits in \(\vec{r}\).

Constraints

• The given qureg must be a density matrix. It is however straightforward to prepare a density matrix from a statevector.

  // let qureg be the intended initial statevector
   
  Qureg temp = createDensityQureg(qureg.numQubits);
  initPureState(temp, qureg);
   
  Qureg reduced = calcReducedDensityMatrix(temp, retainQubits, numRetainQubits);
  destroyQureg(temp);

• When qureg is distributed, the returned Qureg will also be distributed, which imposes a minimum on the number of qubits contained within; \(\log_2(W)\) where \(W\) is the number of distributed nodes (or "world size"). This imposes bounds upon numRetainQubits of

  qureg.logNumNodes <= numRetainQubits <= qureg.numQubits - 1

Equivalences

• The function calcPartialTrace() is entirely equivalent, but permits directly specifying the qubits to be traced out.
• The functions setQuregToPartialTrace() and setQuregToReducedDensityMatrix() are also equivalent but permit overwriting an existing Qureg.

Example

Qureg state = createDensityQureg(5);
initRandomMixedState(state, 10);
reportQureg(state);
 
int qubits[] = {1,3};
Qureg reduced = calcReducedDensityMatrix(state, qubits, 2);
reportQureg(reduced);
 
// state's qubits {1,3} have become reduced's qubits {0,1}

Parameters

[in]	qureg	a density matrix.
[in]	retainQubits	a list of qubits to retain in the reduced density matrix (at shifted, contiguous indices).
[in]	numRetainQubits	the length of retainQubits.

Returns

A new Qureg containing numRetainQubits qubits, initialised to the reduced density matrix of qureg.

Exceptions

error	if qureg is uninitialised. if numRetainQubits is less than one. if numRetainQubits is equal or greater than the number of qubits in qureg. if qureg is distributed and numRetainQubits is less than qureg.logNumNodes. if the system contains insufficient RAM (or VRAM) to store the new Qureg in any deployment. if any memory allocation of the output Qureg unexpectedly fails.
seg-fault	if retainQubits is not a list of length numRetainQubits.

Attention

This function's input validation has not yet been unit tested, so erroneous usage may produce unexpected output. Please use with caution!

See also

• calcPartialTrace()
• setQuregToPartialTrace()
• setQuregToReducedDensityMatrix()

Author

Tyson Jones

Definition at line 394 of file calculations.cpp.

                                                                                    {
    validate_quregFields(qureg, __func__);
    validate_quregIsDensityMatrix(qureg, __func__);
    validate_targets(qureg, retainQubits, numRetainQubits, __func__);
    validate_quregCanBeReduced(qureg, qureg.numQubits - numRetainQubits, __func__);
 
    auto traceQubits = util_getNonTargetedQubits(lists_getList64(retainQubits, numRetainQubits), qureg.numQubits);
 
    // harmlessly re-validates
    return calcPartialTrace(qureg, traceQubits.data(), traceQubits.size());
}

Referenced by calcReducedDensityMatrix(), and TEST_CASE().