Deutsch’s prescription (D-CTCs)#

Functions#

Calculate the chronology-violating (CV) state(s) according to the D-CTC prescription by computing fixed points of the map

(310)#\[\MapDCTCsCV_{\Unitary}[\StateCR,\StateCV] = \trace_\CR\bigl[\Unitary(\StateCR \otimes \StateCV)\Unitary^\dagger\bigr]\]

given the chronology-respecting (CR) input state input_respecting (\(\StateCR\)) and (unitary) interaction described by gate (\(\Unitary\)).

Parameters:

input_respecting (mat | QuantumState) – The matrix representation of the chronology-respecting (CR) input state.
gate (mat | QuantumGate) – The matrix representation of the gate describing the (unitary) interaction between the CR and CV systems.
systems_respecting (list[int]) – The numerical indices of the chronology-respecting (CR) subsystems.
systems_violating (list[int]) – The numerical indices of the chronology-violating (CV) subsystems.
free_symbol (sym | str) – The representation of the algebraic symbol to be used as the free parameter in the case where the CV map has a multiplicity of fixed points. Defaults to "g".

Returns:

mat – The fixed-point solution(s) of the D-CTC CV map.

Note

Please note that this function in its current form is considered to be highly experimental.

dctc_respecting( input_respecting: MutableDenseMatrix | QuantumState, input_violating: MutableDenseMatrix | QuantumState, gate: MutableDenseMatrix | QuantumGate, systems_respecting: list[int], systems_violating: list[int], ) → MutableDenseMatrix[source]#

Calculate the chronology-respecting (CR) state(s) according to the D-CTC prescription’s CR map

(311)#\[\MapDCTCsCR_{\Unitary}[\StateCR,\StateCV] = \trace_\CV\bigl[\Unitary(\StateCR \otimes \StateCV)\Unitary^\dagger\bigr]\]

given the chronology-respecting (CR) input state input_respecting (\(\StateCR\)), chronology-violating (CV) solution state input_violating (\(\StateCV\)), and (unitary) interaction described by gate (\(\Unitary\)).

Parameters:

input_respecting (mat | QuantumState) – The matrix representation of the chronology-respecting (CR) input state.
input_violating (mat | QuantumState) – The matrix representation of the chronology-violating (CR) solution state.
gate (mat | QuantumGate) – The matrix representation of the gate describing the (unitary) interaction between the CR and CV systems.
systems_respecting (list[int]) – The numerical indices of the chronology-respecting (CR) subsystems.
systems_violating (list[int]) – The numerical indices of the chronology-violating (CV) subsystems.

Returns:

mat – The solution(s) of the D-CTC CR map.

Class#

class DCTC(

*args,

free_symbol: MatrixSymbol | MatrixElement | Symbol | str | None = None,

**kwargs,

)[source]#

Bases: QuantumCTC

A subclass for creating closed timelike curves described by Deutsch’s prescription (D-CTCs) of quantum time travel.

This is built upon the QuantumCTC class, and so inherits all of its attributes, properties, and methods.

Parameters:

*args – Variable-length argument list, passed directly to the constructor __init__ of the superclass QuantumGate.
free_symbol (sym | str) – The representation of the algebraic symbol to be used as the free parameter in the case where the CV map has a multiplicity of fixed points. Defaults to "g".
**kwargs – Arbitrary keyword arguments, passed directly to the constructor __init__ of the superclass QuantumGate.

Examples

For usage examples, please see the superclass QuantumCTC.

Constructor argument properties#

property DCTC.free_symbol: MatrixSymbol | MatrixElement | Symbol | str#: The representation of the algebraic symbol to be used as the free parameter in the case where the CV map has a multiplicity of fixed points.

Read-only properties#

property DCTC.matrix: MutableDenseMatrix#: The matrix representation of the total D-CTC chronology-respecting (CR) output state prior to any post-processing.

Methods#

Compute the matrix representation of the D-CTC chronology-violating (CV) state(s).

Parameters:

conditions (list[tuple[num | sym | str, num | sym | str]]) – Algebraic conditions to be applied to the state. Defaults to the value of self.conditions.
simplify (bool) – Whether to perform algebraic simplification on the state. Defaults to False.
conjugate (bool) – Whether to perform Hermitian conjugation on the state. Defaults to False.
free_symbol (str) – The string representation of the algebraic symbol to be used as the free parameter in the case where the CV map has a multiplicity of fixed points. Defaults to the value of self.free_symbol.

Returns:

mat – The matrix representation of the CV output state.

Compute the matrix representation of the D-CTC chronology-respecting (CR) state(s) (including any post-processing).

Parameters:

conditions (list[tuple[num | sym | str, num | sym | str]]) – Algebraic conditions to be applied to the state. Defaults to the value of self.conditions.
simplify (bool) – Whether to perform algebraic simplification on the state. Defaults to False.
conjugate (bool) – Whether to perform Hermitian conjugation on the state. Defaults to False.
postprocess (bool) – Whether to post-process the state (i.e., perform the circuit’s traces and postselections). Defaults to True.
free_symbol (str) – The string representation of the algebraic symbol to be used as the free parameter in the case where the CV map has a multiplicity of fixed points. Defaults to the value of self.free_symbol.

Returns:

mat – The matrix representation of the (post-processed) CR output state.

An alias for the output_respecting() method.

Useful for polymorphism.

Parameters:

conditions (list[tuple[num | sym | str, num | sym | str]]) – Algebraic conditions to be applied to the state. Defaults to the value of self.conditions.
simplify (bool) – Whether to perform algebraic simplification on the state. Defaults to False.
conjugate (bool) – Whether to perform Hermitian conjugation on the state. Defaults to False.
postprocess (bool) – Whether to post-process the state (i.e., perform the circuit’s traces and postselections). Defaults to True.
free_symbol (str) – The string representation of the algebraic symbol to be used as the free parameter in the case where the CV map has a multiplicity of fixed points. Defaults to the value of self.free_symbol.

Returns:

mat – The matrix representation of the (post-processed) CR output state.

Compute the D-CTC chronology-violating (CV) state(s) as a QuantumState instance.

Parameters:

conditions (list[tuple[num | sym | str, num | sym | str]]) – Algebraic conditions to be applied to the state. Defaults to the value of self.conditions.
simplify (bool) – Whether to perform algebraic simplification on the state before committing it to the matrix property. Defaults to False.
conjugate (bool) – Whether to perform Hermitian conjugation on the state. Defaults to False.
norm (bool | num | sym | str) – The value to which the state is normalized. If True, normalizes to a value of \(1\). If False, does not normalize. Defaults to False.
label (str) – The unformatted string used to represent the state in mathematical expressions. Must have a non-zero length. Defaults to "ρ" (if form == "matrix") or "ψ" (if form == "vector").
notation (str) – The formatted string used to represent the state in mathematical expressions. When not None, overrides the value passed to label. Must have a non-zero length. Not intended to be set by the user in most cases. Defaults to None.
traces (list[int]) – A list of indices of the CV systems (relative to the entire circuit) on which to perform partial traces. Defaults to [].
free_symbol (str) – The string representation of the algebraic symbol to be used as the free parameter in the case where the CV map has a multiplicity of fixed points. Defaults to the value of self.free_symbol.
debug (bool) – Whether to print the internal state (held in matrix) on change. Defaults to False.

Returns:

QuantumState – The CV output state as a QuantumState instance.

Compute the D-CTC chronology-respecting (CR) state(s) as a QuantumState instance.

Parameters:

conditions (list[tuple[num | sym | str, num | sym | str]]) – Algebraic conditions to be applied to the state. Defaults to the value of self.conditions.
simplify (bool) – Whether to perform algebraic simplification on the state before committing it to the matrix property. Defaults to False.
conjugate (bool) – Whether to perform Hermitian conjugation on the state. Defaults to False.
norm (bool | num | sym | str) – The value to which the state is normalized. If True, normalizes to a value of \(1\). If False, does not normalize. Defaults to False.
label (str) – The unformatted string used to represent the state in mathematical expressions. Must have a non-zero length. Defaults to "ρ" (if form == "matrix") or "ψ" (if form == "vector").
notation (str) – The formatted string used to represent the state in mathematical expressions. When not None, overrides the value passed to label. Must have a non-zero length. Not intended to be set by the user in most cases. Defaults to None.
traces (list[int]) – A list of indices of the CR systems (relative to the entire circuit) on which to perform partial traces. Performed regardless of the value of postprocess. Defaults to [].
postprocess (bool) – Whether to post-process the state (i.e., perform the circuit’s traces and postselections). Defaults to True.
free_symbol (str) – The string representation of the algebraic symbol to be used as the free parameter in the case where the CV map has a multiplicity of fixed points. Defaults to the value of self.free_symbol.
debug (bool) – Whether to print the internal state (held in matrix) on change. Defaults to False.

Returns:

QuantumState – The (post-processed) CR output state as a QuantumState instance.