density

package
v0.0.3 Latest Latest
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 Go to latest Published: Sep 9, 2023 License: MIT Imports: 7 Imported by: 0

Documentation

Index

Examples

Constants

This section is empty.

Variables

This section is empty.

Functions

func BitFlip 

func BitFlip(p float64) (matrix.Matrix, matrix.Matrix, error)

BitFlip returns the bit flip channel.

Example 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
)

func main() {
	m0, m1, _ := density.BitFlip(0.5)

	for _, r := range m0 {
		fmt.Println(r)
	}
	fmt.Println()

	for _, r := range m1 {
		fmt.Println(r)
	}

}

Output:

[(0.7071067811865476+0i) (0+0i)]
[(0+0i) (0.7071067811865476+0i)]

[(0+0i) (0.7071067811865476+0i)]
[(0.7071067811865476+0i) (0+0i)]

func BitPhaseFlip 

func BitPhaseFlip(p float64) (matrix.Matrix, matrix.Matrix, error)

BitPhaseFlip returns the bit-phase flip channel.

Example 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
)

func main() {
	m0, m1, _ := density.BitPhaseFlip(0.5)

	for _, r := range m0 {
		fmt.Println(r)
	}
	fmt.Println()

	for _, r := range m1 {
		fmt.Println(r)
	}

}

Output:

[(0.7071067811865476+0i) (0+0i)]
[(0+0i) (0.7071067811865476+0i)]

[(0+0i) (0-0.7071067811865476i)]
[(0+0.7071067811865476i) (0+0i)]

func Flip 

func Flip(p float64, m matrix.Matrix) (matrix.Matrix, matrix.Matrix, error)

Flip returns the flip channel.

func PhaseFlip 

func PhaseFlip(p float64) (matrix.Matrix, matrix.Matrix, error)

PhaseFlip returns the phase flip channel.

Example 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
)

func main() {
	m0, m1, _ := density.PhaseFlip(0.5)

	for _, r := range m0 {
		fmt.Println(r)
	}
	fmt.Println()

	for _, r := range m1 {
		fmt.Println(r)
	}

}

Output:

[(0.7071067811865476+0i) (0+0i)]
[(0+0i) (0.7071067811865476+0i)]

[(0.7071067811865476+0i) (0+0i)]
[(0+0i) (-0.7071067811865476+0i)]

Types

type Matrix 

type Matrix struct {
	// contains filtered or unexported fields
}

Matrix is a density matrix.

func New 

func New(ensemble []State) (*Matrix, error)

New returns a new density matrix.

func (*Matrix) Add 

func (m *Matrix) Add(ensemble []State) error

Add adds a quantum state to the density matrix.

func (*Matrix) Apply 

func (m *Matrix) Apply(u matrix.Matrix) *Matrix

Apply applies a unitary matrix to the density matrix.

func (*Matrix) Depolarizing 

func (m *Matrix) Depolarizing(p float64) (*Matrix, error)

Depolarizing returns the depolarizing channel.

Example 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
	"github.com/itsubaki/q/quantum/qubit"
)

func main() {
	rho, _ := density.New([]density.State{{1.0, qubit.Zero()}})
	fmt.Printf("0: %.2f\n", rho.Measure(qubit.Zero()))
	fmt.Printf("1: %.2f\n", rho.Measure(qubit.One()))
	fmt.Println()

	dep, _ := rho.Depolarizing(1)
	fmt.Printf("0: %.2f\n", dep.Measure(qubit.Zero()))
	fmt.Printf("1: %.2f\n", dep.Measure(qubit.One()))

}

Output:

0: 1.00
1: 0.00

0: 0.50
1: 0.50

func (*Matrix) Dimension 

func (m *Matrix) Dimension() (int, int)

Dimension returns the dimension of the density matrix.

func (*Matrix) ExpectedValue 

func (m *Matrix) ExpectedValue(u matrix.Matrix) float64

ExpectationValue returns the expectation value of the given operator.

Example 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
	"github.com/itsubaki/q/quantum/gate"
	"github.com/itsubaki/q/quantum/qubit"
)

func main() {
	rho, _ := density.New([]density.State{
		{0.1, qubit.Zero()},
		{0.9, qubit.Zero().Apply(gate.H())},
	})

	fmt.Printf("X: %.2v\n", rho.ExpectedValue(gate.X()))
	fmt.Printf("Y: %.2v\n", rho.ExpectedValue(gate.Y()))
	fmt.Printf("Z: %.2v\n", rho.ExpectedValue(gate.Z()))

}

Output:

X: 0.9
Y: 0
Z: 0.1

func (*Matrix) Measure 

func (m *Matrix) Measure(q *qubit.Qubit) float64

Measure returns the probability of measuring the qubit in the given state.

Example 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
	"github.com/itsubaki/q/quantum/qubit"
)

func main() {
	rho, _ := density.New([]density.State{
		{0.1, qubit.Zero()},
		{0.9, qubit.One()},
	})

	fmt.Printf("0: %.2v\n", rho.Measure(qubit.Zero()))
	fmt.Printf("1: %.2v\n", rho.Measure(qubit.One()))

}

Output:

0: 0.1
1: 0.9

func (*Matrix) NumberOfBit 

func (m *Matrix) NumberOfBit() int

NumberOfBit returns the number of qubits.

func (*Matrix) PartialTrace 

func (m *Matrix) PartialTrace(index ...int) *Matrix

PartialTrace returns the partial trace of the density matrix.

Example 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
	"github.com/itsubaki/q/quantum/gate"
	"github.com/itsubaki/q/quantum/qubit"
)

func main() {
	rho, _ := density.New([]density.State{
		{0.5, qubit.Zero(2).Apply(gate.QFT(2))},
		{0.5, qubit.One(2).Apply(gate.QFT(2))},
	})

	for _, r := range rho.Raw() {
		fmt.Printf("%.4f\n", r)
	}
	fmt.Printf("trace: %.2v, square_trace: %.2v\n\n", rho.Trace(), rho.SquareTrace())

	p0 := rho.PartialTrace(0)
	for _, r := range p0.Raw() {
		fmt.Printf("%.4f\n", r)
	}
	fmt.Printf("trace: %.2v, square_trace: %.2v\n\n", p0.Trace(), p0.SquareTrace())

	p1 := rho.PartialTrace(1)
	for _, r := range p1.Raw() {
		fmt.Printf("%.4f\n", r)
	}
	fmt.Printf("trace: %.2v, square_trace: %.2v\n\n", p1.Trace(), p1.SquareTrace())

}

Output:

[(0.2500+0.0000i) (0.0000+0.0000i) (0.1250+0.1250i) (0.1250-0.1250i)]
[(0.0000+0.0000i) (0.2500+0.0000i) (0.1250-0.1250i) (0.1250+0.1250i)]
[(0.1250-0.1250i) (0.1250+0.1250i) (0.2500+0.0000i) (0.0000+0.0000i)]
[(0.1250+0.1250i) (0.1250-0.1250i) (0.0000+0.0000i) (0.2500+0.0000i)]
trace: 1, square_trace: 0.5

[(0.5000+0.0000i) (0.0000+0.0000i)]
[(0.0000+0.0000i) (0.5000+0.0000i)]
trace: 1, square_trace: 0.5

[(0.5000+0.0000i) (0.2500+0.2500i)]
[(0.2500-0.2500i) (0.5000+0.0000i)]
trace: 1, square_trace: 0.75
Example (X16) 
package main

import (
	"fmt"

	"github.com/itsubaki/q/math/matrix"
	"github.com/itsubaki/q/quantum/density"
	"github.com/itsubaki/q/quantum/gate"
	"github.com/itsubaki/q/quantum/qubit"
)

func main() {
	rho, _ := density.New([]density.State{
		{1.0, qubit.Zero(4).Apply(matrix.TensorProduct(gate.H(2), gate.X(), gate.Z()).Apply(gate.CNOT(4, 1, 3)).Apply(gate.CNOT(4, 0, 2)))},
	})

	p01 := rho.PartialTrace(0, 1)
	fmt.Printf("trace: %.2f\n", p01.Trace())
	fmt.Printf("square_trace: %.2f\n", p01.SquareTrace())

}

Output:

trace: 1.00
square_trace: 0.25
Example (X8) 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
	"github.com/itsubaki/q/quantum/gate"
	"github.com/itsubaki/q/quantum/qubit"
)

func main() {
	rho, _ := density.New([]density.State{
		{0.5, qubit.Zero(3).Apply(gate.QFT(3))},
		{0.5, qubit.One(3).Apply(gate.QFT(3))},
	})

	for _, r := range rho.Raw() {
		fmt.Printf("%.4f\n", r)
	}
	fmt.Printf("trace: %.2v, square_trace: %.2v\n\n", rho.Trace(), rho.SquareTrace())

	p0 := rho.PartialTrace(0)
	for _, r := range p0.Raw() {
		fmt.Printf("%.4f\n", r)
	}
	fmt.Printf("trace: %.2v, square_trace: %.2v\n\n", p0.Trace(), p0.SquareTrace())

	p1 := rho.PartialTrace(1)
	for _, r := range p1.Raw() {
		fmt.Printf("%.4f\n", r)
	}
	fmt.Printf("trace: %.2v, square_trace: %.2v\n\n", p1.Trace(), p1.SquareTrace())

	p2 := rho.PartialTrace(2)
	for _, r := range p2.Raw() {
		fmt.Printf("%.4f\n", r)
	}
	fmt.Printf("trace: %.2v, square_trace: %.2v\n\n", p2.Trace(), p2.SquareTrace())

}

Output:

[(0.1250+0.0000i) (0.0000+0.0000i) (0.0625+0.0625i) (0.0625-0.0625i) (0.1067+0.0442i) (0.0183-0.0442i) (0.0183+0.0442i) (0.1067-0.0442i)]
[(0.0000+0.0000i) (0.1250+0.0000i) (0.0625-0.0625i) (0.0625+0.0625i) (0.0183-0.0442i) (0.1067+0.0442i) (0.1067-0.0442i) (0.0183+0.0442i)]
[(0.0625-0.0625i) (0.0625+0.0625i) (0.1250+0.0000i) (0.0000+0.0000i) (0.1067-0.0442i) (0.0183+0.0442i) (0.1067+0.0442i) (0.0183-0.0442i)]
[(0.0625+0.0625i) (0.0625-0.0625i) (0.0000+0.0000i) (0.1250+0.0000i) (0.0183+0.0442i) (0.1067-0.0442i) (0.0183-0.0442i) (0.1067+0.0442i)]
[(0.1067-0.0442i) (0.0183+0.0442i) (0.1067+0.0442i) (0.0183-0.0442i) (0.1250+0.0000i) (0.0000+0.0000i) (0.0625+0.0625i) (0.0625-0.0625i)]
[(0.0183+0.0442i) (0.1067-0.0442i) (0.0183-0.0442i) (0.1067+0.0442i) (0.0000+0.0000i) (0.1250+0.0000i) (0.0625-0.0625i) (0.0625+0.0625i)]
[(0.0183-0.0442i) (0.1067+0.0442i) (0.1067-0.0442i) (0.0183+0.0442i) (0.0625-0.0625i) (0.0625+0.0625i) (0.1250+0.0000i) (0.0000+0.0000i)]
[(0.1067+0.0442i) (0.0183-0.0442i) (0.0183+0.0442i) (0.1067-0.0442i) (0.0625+0.0625i) (0.0625-0.0625i) (0.0000+0.0000i) (0.1250+0.0000i)]
trace: 1, square_trace: 0.5

[(0.2500+0.0000i) (0.0000+0.0000i) (0.1250+0.1250i) (0.1250-0.1250i)]
[(0.0000+0.0000i) (0.2500+0.0000i) (0.1250-0.1250i) (0.1250+0.1250i)]
[(0.1250-0.1250i) (0.1250+0.1250i) (0.2500+0.0000i) (0.0000+0.0000i)]
[(0.1250+0.1250i) (0.1250-0.1250i) (0.0000+0.0000i) (0.2500+0.0000i)]
trace: 1, square_trace: 0.5

[(0.2500+0.0000i) (0.0000+0.0000i) (0.2134+0.0884i) (0.0366-0.0884i)]
[(0.0000+0.0000i) (0.2500+0.0000i) (0.0366-0.0884i) (0.2134+0.0884i)]
[(0.2134-0.0884i) (0.0366+0.0884i) (0.2500+0.0000i) (0.0000+0.0000i)]
[(0.0366+0.0884i) (0.2134-0.0884i) (0.0000+0.0000i) (0.2500+0.0000i)]
trace: 1, square_trace: 0.5

[(0.2500+0.0000i) (0.1250+0.1250i) (0.2134+0.0884i) (0.0366+0.0884i)]
[(0.1250-0.1250i) (0.2500+0.0000i) (0.2134-0.0884i) (0.2134+0.0884i)]
[(0.2134-0.0884i) (0.2134+0.0884i) (0.2500+0.0000i) (0.1250+0.1250i)]
[(0.0366-0.0884i) (0.2134-0.0884i) (0.1250-0.1250i) (0.2500+0.0000i)]
trace: 1, square_trace: 0.71

func (*Matrix) Raw 

func (m *Matrix) Raw() matrix.Matrix

Raw returns the raw matrix.

func (*Matrix) SquareTrace 

func (m *Matrix) SquareTrace() float64

SquareTrace returns the square trace of the density matrix.

Example 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
	"github.com/itsubaki/q/quantum/qubit"
)

func main() {
	pure, _ := density.New([]density.State{{1.0, qubit.Zero()}})
	mixed, _ := density.New([]density.State{{0.1, qubit.Zero()}, {0.9, qubit.One()}})

	fmt.Printf("pure:  %.2f\n", pure.SquareTrace())
	fmt.Printf("mixed: %.2f\n", mixed.SquareTrace())

}

Output:

pure:  1.00
mixed: 0.82

func (*Matrix) Trace 

func (m *Matrix) Trace() float64

Trace returns the trace of the density matrix.

Example 
package main

import (
	"fmt"

	"github.com/itsubaki/q/quantum/density"
	"github.com/itsubaki/q/quantum/qubit"
)

func main() {
	pure, _ := density.New([]density.State{{1.0, qubit.Zero()}})
	mixed, _ := density.New([]density.State{{0.1, qubit.Zero()}, {0.9, qubit.One()}})

	fmt.Printf("pure:  %.2f\n", pure.Trace())
	fmt.Printf("mixed: %.2f\n", mixed.Trace())

}

Output:

pure:  1.00
mixed: 1.00

type State 

type State struct {
	Probability float64
	Qubit       *qubit.Qubit
}

State is a quantum state.

Source Files

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