Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z :...

82
Exploring Quantum Chaos with Quantum Information Processors David Poulin Institute for Quantum Computing Perimeter Institute for Theoretical Physics Caltech, November 2004 – p.1

Transcript of Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z :...

Page 1: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Exploring Quantum Chaos withQuantum Information Processors

David Poulin

Institute for Quantum Computing

Perimeter Institute for Theoretical Physics

Caltech, November 2004 – p.1

Page 2: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

OutlineSimulating quantum systems: the readout problem.

The QDC1 model of computation.

The elementary scattering circuit.

Looking for symmetries in complex quantum systems.Random matrix conjecture.Estimating the form factors.

Hypersensitivity to perturbations: fidelity decay.DQC1 algorithm.Experimental results.Decoherence and dynamical instability.Entanglement, decoherence and quantum-computationalspeed-up.

Conclusion.

Caltech, November 2004 – p.2

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Readout problem

In a physical experiment, a system evolves from its initial state andis finally measured in some basis.

We know that quantum computer can be use to simulate theevolution of some quantum systems (Zalka, Lloyd, ...).

U(t) ≈ Un . . . U2U1︸ ︷︷ ︸

Universal set

Do we need to prepare a special initial state?

Low temperature transport properties 〈ψ0|e−i(H0+V )|ψk〉.

Do we need to know the spectral projectors of the measuredquantity?

The ability to efficiently simulate the dynamics U(t) alone can beused to extract interesting physical quantities.

Caltech, November 2004 – p.3

Page 4: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Readout problem

In a physical experiment, a system evolves from its initial state andis finally measured in some basis.

We know that quantum computer can be use to simulate theevolution of some quantum systems (Zalka, Lloyd, ...).

U(t) ≈ Un . . . U2U1︸ ︷︷ ︸

Universal set

Do we need to prepare a special initial state?

Low temperature transport properties 〈ψ0|e−i(H0+V )|ψk〉.

Do we need to know the spectral projectors of the measuredquantity?

The ability to efficiently simulate the dynamics U(t) alone can beused to extract interesting physical quantities.

Caltech, November 2004 – p.3

Page 5: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Readout problem

In a physical experiment, a system evolves from its initial state andis finally measured in some basis.

We know that quantum computer can be use to simulate theevolution of some quantum systems (Zalka, Lloyd, ...).

U(t) ≈ Un . . . U2U1︸ ︷︷ ︸

Universal set

Do we need to prepare a special initial state?

Low temperature transport properties 〈ψ0|e−i(H0+V )|ψk〉.

Do we need to know the spectral projectors of the measuredquantity?

The ability to efficiently simulate the dynamics U(t) alone can beused to extract interesting physical quantities.

Caltech, November 2004 – p.3

Page 6: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Readout problem

In a physical experiment, a system evolves from its initial state andis finally measured in some basis.

We know that quantum computer can be use to simulate theevolution of some quantum systems (Zalka, Lloyd, ...).

U(t) ≈ Un . . . U2U1︸ ︷︷ ︸

Universal set

Do we need to prepare a special initial state?

Low temperature transport properties 〈ψ0|e−i(H0+V )|ψk〉.Do we need to know the spectral projectors of the measuredquantity?

The ability to efficiently simulate the dynamics U(t) alone can beused to extract interesting physical quantities.

Caltech, November 2004 – p.3

Page 7: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Readout problem

In a physical experiment, a system evolves from its initial state andis finally measured in some basis.

We know that quantum computer can be use to simulate theevolution of some quantum systems (Zalka, Lloyd, ...).

U(t) ≈ Un . . . U2U1︸ ︷︷ ︸

Universal set

Do we need to prepare a special initial state?

Low temperature transport properties 〈ψ0|e−i(H0+V )|ψk〉.Do we need to know the spectral projectors of the measuredquantity?

The ability to efficiently simulate the dynamics U(t) alone can beused to extract interesting physical quantities.

Caltech, November 2004 – p.3

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DQC1

Liquid state NMR QIP

The initial state is thermal ρ = 1Z e

−βH .

H '∑

i ~µi · ~B +∑

i6=j Jij~µi~µj , ω � J and β � 1, so

ρ ' 1

Z

(

1l− β∑

i

ωiσzi

)

Apply an inhomogeneous magnetic field to induce a randomphase:

ρ→ 1

Z

(

1l +βωn

2n|0 . . . 00〉〈0 . . . 00|

)

Pseudo pure state, the identity part does not contribute to thecomputation.D.G. Cory, A.F. Fahmy, and T.F. Havel, 1997, & N.A. Gershenfeld and I.L. Chuang, 1997.

Caltech, November 2004 – p.4

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DQC1

Liquid state NMR QIP

The initial state is thermal ρ = 1Z e

−βH .

H '∑

i ~µi · ~B +∑

i6=j Jij~µi~µj , ω � J and β � 1, so

ρ ' 1

Z

(

1l− β∑

i

ωiσzi

)

Apply an inhomogeneous magnetic field to induce a randomphase:

ρ→ 1

Z

(

1l +βωn

2n|0 . . . 00〉〈0 . . . 00|

)

Pseudo pure state, the identity part does not contribute to thecomputation.D.G. Cory, A.F. Fahmy, and T.F. Havel, 1997, & N.A. Gershenfeld and I.L. Chuang, 1997.

Caltech, November 2004 – p.4

Page 10: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

DQC1

Liquid state NMR QIP

The initial state is thermal ρ = 1Z e

−βH .

H '∑

i ~µi · ~B +∑

i6=j Jij~µi~µj , ω � J and β � 1, so

ρ ' 1

Z

(

1l− β∑

i

ωiσzi

)

Apply an inhomogeneous magnetic field to induce a randomphase:

ρ→ 1

Z

(

1l +βωn

2n|0 . . . 00〉〈0 . . . 00|

)

Pseudo pure state, the identity part does not contribute to thecomputation.D.G. Cory, A.F. Fahmy, and T.F. Havel, 1997, & N.A. Gershenfeld and I.L. Chuang, 1997.

Caltech, November 2004 – p.4

Page 11: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

DQC1

Liquid state NMR QIP

The initial state is thermal ρ = 1Z e

−βH .

H '∑

i ~µi · ~B +∑

i6=j Jij~µi~µj , ω � J and β � 1, so

ρ ' 1

Z

(

1l− β∑

i

ωiσzi

)

Apply an inhomogeneous magnetic field to induce a randomphase:

ρ→ 1

Z

(

1l +βωn

2n|0 . . . 00〉〈0 . . . 00|

)

Pseudo pure state, the identity part does not contribute to thecomputation.D.G. Cory, A.F. Fahmy, and T.F. Havel, 1997, & N.A. Gershenfeld and I.L. Chuang, 1997.

Caltech, November 2004 – p.4

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DQC1

This exponential decay can be avoided using algorithmiccooling, but the overhead is ' 1010.

Add restrictions to the “standard” model of quantumcomputation1. Only a single pseudo-pure qubit.

ρ =

(1− ε

21l + ε|0〉〈0|

)

⊗ 1

2n1l

2. No projective measurement, rather 〈σz〉 within finiteaccuracy µ.

Can we do something non trivial with this?E. Knill and R. Laflamme, 1998

Caltech, November 2004 – p.5

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DQC1

This exponential decay can be avoided using algorithmiccooling, but the overhead is ' 1010.

Add restrictions to the “standard” model of quantumcomputation

1. Only a single pseudo-pure qubit.

ρ =

(1− ε

21l + ε|0〉〈0|

)

⊗ 1

2n1l

2. No projective measurement, rather 〈σz〉 within finiteaccuracy µ.

Can we do something non trivial with this?E. Knill and R. Laflamme, 1998

Caltech, November 2004 – p.5

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DQC1

This exponential decay can be avoided using algorithmiccooling, but the overhead is ' 1010.

Add restrictions to the “standard” model of quantumcomputation1. Only a single pseudo-pure qubit.

ρ =

(1− ε

21l + ε|0〉〈0|

)

⊗ 1

2n1l

2. No projective measurement, rather 〈σz〉 within finiteaccuracy µ.

Can we do something non trivial with this?E. Knill and R. Laflamme, 1998

Caltech, November 2004 – p.5

Page 15: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

DQC1

This exponential decay can be avoided using algorithmiccooling, but the overhead is ' 1010.

Add restrictions to the “standard” model of quantumcomputation1. Only a single pseudo-pure qubit.

ρ =

(1− ε

21l + ε|0〉〈0|

)

⊗ 1

2n1l

2. No projective measurement, rather 〈σz〉 within finiteaccuracy µ.

Can we do something non trivial with this?E. Knill and R. Laflamme, 1998

Caltech, November 2004 – p.5

Page 16: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

DQC1

This exponential decay can be avoided using algorithmiccooling, but the overhead is ' 1010.

Add restrictions to the “standard” model of quantumcomputation1. Only a single pseudo-pure qubit.

ρ =

(1− ε

21l + ε|0〉〈0|

)

⊗ 1

2n1l

2. No projective measurement, rather 〈σz〉 within finiteaccuracy µ.

Can we do something non trivial with this?E. Knill and R. Laflamme, 1998

Caltech, November 2004 – p.5

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DQC1

Remarks:

1. Not robust against noise.Nowhere to dump the entropy!

2. Measurement accuracy µ→ δ: cost (µ/δ)2.

3. Pseudo-pure state ⇔ pure state ρ = |0〉〈0| ⊗ 12n 1l: cost 1/ε2.

4. One pure qubit ⇔ log(n) pure qubits: cost poly(n).From a computational complexity point of view:

|00 . . . 0〉〈00 . . . 0|︸ ︷︷ ︸

ln n

⊗ 1

2n1l

Caltech, November 2004 – p.6

Page 18: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

DQC1

Remarks:

1. Not robust against noise.Nowhere to dump the entropy!

2. Measurement accuracy µ→ δ: cost (µ/δ)2.

3. Pseudo-pure state ⇔ pure state ρ = |0〉〈0| ⊗ 12n 1l: cost 1/ε2.

4. One pure qubit ⇔ log(n) pure qubits: cost poly(n).From a computational complexity point of view:

|00 . . . 0〉〈00 . . . 0|︸ ︷︷ ︸

ln n

⊗ 1

2n1l

Caltech, November 2004 – p.6

Page 19: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

DQC1

Remarks:

1. Not robust against noise.Nowhere to dump the entropy!

2. Measurement accuracy µ→ δ: cost (µ/δ)2.

3. Pseudo-pure state ⇔ pure state ρ = |0〉〈0| ⊗ 12n 1l: cost 1/ε2.

4. One pure qubit ⇔ log(n) pure qubits: cost poly(n).From a computational complexity point of view:

|00 . . . 0〉〈00 . . . 0|︸ ︷︷ ︸

ln n

⊗ 1

2n1l

Caltech, November 2004 – p.6

Page 20: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

DQC1

Remarks:

1. Not robust against noise.Nowhere to dump the entropy!

2. Measurement accuracy µ→ δ: cost (µ/δ)2.

3. Pseudo-pure state ⇔ pure state ρ = |0〉〈0| ⊗ 12n 1l: cost 1/ε2.

4. One pure qubit ⇔ log(n) pure qubits: cost poly(n).From a computational complexity point of view:

|00 . . . 0〉〈00 . . . 0|︸ ︷︷ ︸

ln n

⊗ 1

2n1l

Caltech, November 2004 – p.6

Page 21: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Scattering circuit

r

U

H H 〈σk〉|0〉〈0|

ργ = Tr{Uρ}

ρ′

1=

1

2

(1 + Re{γ} iIm{γ}−iIm{γ} 1−Re{γ}

)

k = z : 〈σz〉 = Tr{ρ′

1σz} = Re{γ} = Re [Tr{ρU}]

k = y : 〈σy〉 = Tr{ρ′

1σy} = Im{γ} = Im [Tr{ρU}]

If U can be implemented efficiently and ρ prepared efficiently, wecan evaluate Tr{ρU} within accuracy δ in time poly(n)/δ2.

Caltech, November 2004 – p.7

Page 22: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Scattering circuit

r

U

H H 〈σk〉|0〉〈0|

ργ = Tr{Uρ}

ρ′

1=

1

2

(1 + Re{γ} iIm{γ}−iIm{γ} 1−Re{γ}

)

k = z : 〈σz〉 = Tr{ρ′

1σz} = Re{γ} = Re [Tr{ρU}]

k = y : 〈σy〉 = Tr{ρ′

1σy} = Im{γ} = Im [Tr{ρU}]

If U can be implemented efficiently and ρ prepared efficiently, wecan evaluate Tr{ρU} within accuracy δ in time poly(n)/δ2.

Caltech, November 2004 – p.7

Page 23: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Scattering circuit

r

U

H H 〈σk〉|0〉〈0|

ργ = Tr{Uρ}

ρ′

1=

1

2

(1 + Re{γ} iIm{γ}−iIm{γ} 1−Re{γ}

)

k = z : 〈σz〉 = Tr{ρ′

1σz} = Re{γ} = Re [Tr{ρU}]

k = y : 〈σy〉 = Tr{ρ′

1σy} = Im{γ} = Im [Tr{ρU}]

If U can be implemented efficiently and ρ prepared efficiently, wecan evaluate Tr{ρU} within accuracy δ in time poly(n)/δ2.

Caltech, November 2004 – p.7

Page 24: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Scattering circuit

r

U

H H 〈σk〉|0〉〈0|

ργ = Tr{Uρ}

ρ′

1=

1

2

(1 + Re{γ} iIm{γ}−iIm{γ} 1−Re{γ}

)

k = z : 〈σz〉 = Tr{ρ′

1σz} = Re{γ} = Re [Tr{ρU}]

k = y : 〈σy〉 = Tr{ρ′

1σy} = Im{γ} = Im [Tr{ρU}]

If U can be implemented efficiently and ρ prepared efficiently, wecan evaluate Tr{ρU} within accuracy δ in time poly(n)/δ2.

Caltech, November 2004 – p.7

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Scattering circuit

We can evaluate Tr{ρU}

Control U and learn about ρ: quantum state tomography.

Control ρ and learn about U : quantum process tomography.C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002

These algorithms require an exponential repetition of thescattering circuit.

Can we get useful partial information about U?

DQC1 imposes restrictions on ρ. We will focus on ρ ∝ 1l.

γ =1

NTr{U} =

1

N

j

λj

Caltech, November 2004 – p.8

Page 26: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Scattering circuit

We can evaluate Tr{ρU}

Control U and learn about ρ: quantum state tomography.

Control ρ and learn about U : quantum process tomography.C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002

These algorithms require an exponential repetition of thescattering circuit.

Can we get useful partial information about U?

DQC1 imposes restrictions on ρ. We will focus on ρ ∝ 1l.

γ =1

NTr{U} =

1

N

j

λj

Caltech, November 2004 – p.8

Page 27: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Scattering circuit

We can evaluate Tr{ρU}

Control U and learn about ρ: quantum state tomography.

Control ρ and learn about U : quantum process tomography.C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002

These algorithms require an exponential repetition of thescattering circuit.

Can we get useful partial information about U?

DQC1 imposes restrictions on ρ. We will focus on ρ ∝ 1l.

γ =1

NTr{U} =

1

N

j

λj

Caltech, November 2004 – p.8

Page 28: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Scattering circuit

We can evaluate Tr{ρU}

Control U and learn about ρ: quantum state tomography.

Control ρ and learn about U : quantum process tomography.C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002

These algorithms require an exponential repetition of thescattering circuit.

Can we get useful partial information about U?

DQC1 imposes restrictions on ρ. We will focus on ρ ∝ 1l.

γ =1

NTr{U} =

1

N

j

λj

Caltech, November 2004 – p.8

Page 29: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Scattering circuit

We can evaluate Tr{ρU}

Control U and learn about ρ: quantum state tomography.

Control ρ and learn about U : quantum process tomography.C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002

These algorithms require an exponential repetition of thescattering circuit.

Can we get useful partial information about U?

DQC1 imposes restrictions on ρ. We will focus on ρ ∝ 1l.

γ =1

NTr{U} =

1

N

j

λj

Caltech, November 2004 – p.8

Page 30: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Scattering circuit

We can evaluate Tr{ρU}

Control U and learn about ρ: quantum state tomography.

Control ρ and learn about U : quantum process tomography.C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002

These algorithms require an exponential repetition of thescattering circuit.

Can we get useful partial information about U?

DQC1 imposes restrictions on ρ. We will focus on ρ ∝ 1l.

γ =1

NTr{U} =

1

N

j

λj

Caltech, November 2004 – p.8

Page 31: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Doing the obvious thing

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

Is there an intuitive way of understanding these two differentbehaviors?

Can we learn something useful about the dynamics of thesystem with this simple algorithm?

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.9

Page 32: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Doing the obvious thing

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

Is there an intuitive way of understanding these two differentbehaviors?

Can we learn something useful about the dynamics of thesystem with this simple algorithm?

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.9

Page 33: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Doing the obvious thing

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

Is there an intuitive way of understanding these two differentbehaviors?

Can we learn something useful about the dynamics of thesystem with this simple algorithm?

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.9

Page 34: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Doing the obvious thing

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

Is there an intuitive way of understanding these two differentbehaviors?

Can we learn something useful about the dynamics of thesystem with this simple algorithm?

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.9

Page 35: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Doing the obvious thing

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

Is there an intuitive way of understanding these two differentbehaviors?

Can we learn something useful about the dynamics of thesystem with this simple algorithm?

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.9

Page 36: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Doing the obvious thing

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

Is there an intuitive way of understanding these two differentbehaviors?

Can we learn something useful about the dynamics of thesystem with this simple algorithm?

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.9

Page 37: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Doing the obvious thing

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

Is there an intuitive way of understanding these two differentbehaviors?

Can we learn something useful about the dynamics of thesystem with this simple algorithm?

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.9

Page 38: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Doing the obvious thing

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

Is there an intuitive way of understanding these two differentbehaviors?

Can we learn something useful about the dynamics of thesystem with this simple algorithm?

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.9

Page 39: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Quantum control

Quantum information processing is the art of manipulatingcomplex quantum systems.

Control a quantum system = exploit its symmetries (QECC,NSS, bang-bang)

Does my system possess any symmetries?

Systems with as many symmetries as degrees of freedom areintegrable.

These are good candidates for quantum informationprocessors.

Chaotic systems possess no or very few symmetries.

They are hard to control.

They are dynamically unstable (sensitive to perturbations).

Caltech, November 2004 – p.10

Page 40: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Quantum control

Quantum information processing is the art of manipulatingcomplex quantum systems.

Control a quantum system = exploit its symmetries (QECC,NSS, bang-bang)

Does my system possess any symmetries?

Systems with as many symmetries as degrees of freedom areintegrable.

These are good candidates for quantum informationprocessors.

Chaotic systems possess no or very few symmetries.

They are hard to control.

They are dynamically unstable (sensitive to perturbations).

Caltech, November 2004 – p.10

Page 41: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Quantum control

Quantum information processing is the art of manipulatingcomplex quantum systems.

Control a quantum system = exploit its symmetries (QECC,NSS, bang-bang)

Does my system possess any symmetries?

Systems with as many symmetries as degrees of freedom areintegrable.

These are good candidates for quantum informationprocessors.

Chaotic systems possess no or very few symmetries.

They are hard to control.

They are dynamically unstable (sensitive to perturbations).

Caltech, November 2004 – p.10

Page 42: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Quantum control

Quantum information processing is the art of manipulatingcomplex quantum systems.

Control a quantum system = exploit its symmetries (QECC,NSS, bang-bang)

Does my system possess any symmetries?

Systems with as many symmetries as degrees of freedom areintegrable.

These are good candidates for quantum informationprocessors.

Chaotic systems possess no or very few symmetries.

They are hard to control.

They are dynamically unstable (sensitive to perturbations).

Caltech, November 2004 – p.10

Page 43: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Quantum control

Quantum information processing is the art of manipulatingcomplex quantum systems.

Control a quantum system = exploit its symmetries (QECC,NSS, bang-bang)

Does my system possess any symmetries?

Systems with as many symmetries as degrees of freedom areintegrable.

These are good candidates for quantum informationprocessors.

Chaotic systems possess no or very few symmetries.

They are hard to control.

They are dynamically unstable (sensitive to perturbations).

Caltech, November 2004 – p.10

Page 44: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

Resonances of Thorium 232Garg et al., Phys. Rev. 134, B85, 1964.

Caltech, November 2004 – p.11

Page 45: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetriesList of resonances

Level spacing

Caltech, November 2004 – p.12

Page 46: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

Wigner’s intuition: The main characteristics of the spectrum of canbe reproduced by random matrices which possess the samesymmetries as the system.

f(U) '∫f(V )P (V )dV .

P (V ) 6= 0 only for those V with the same symmetries as U .P (V ) is determined with the maximum entropy principlegiven the above constraint.

No symmetry = Level repulsion

U

d

Pr(d)

d

Caltech, November 2004 – p.13

Page 47: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

Wigner’s intuition: The main characteristics of the spectrum of canbe reproduced by random matrices which possess the samesymmetries as the system.

f(U) '∫f(V )P (V )dV .

P (V ) 6= 0 only for those V with the same symmetries as U .P (V ) is determined with the maximum entropy principlegiven the above constraint.

No symmetry = Level repulsion

U

d

Pr(d)

d

Caltech, November 2004 – p.13

Page 48: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

Wigner’s intuition: The main characteristics of the spectrum of canbe reproduced by random matrices which possess the samesymmetries as the system.

f(U) '∫f(V )P (V )dV .

P (V ) 6= 0 only for those V with the same symmetries as U .P (V ) is determined with the maximum entropy principlegiven the above constraint.

No symmetry = Level repulsion

U

d

Pr(d)

d

Caltech, November 2004 – p.13

Page 49: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

Wigner’s intuition: The main characteristics of the spectrum of canbe reproduced by random matrices which possess the samesymmetries as the system.

f(U) '∫f(V )P (V )dV .

P (V ) 6= 0 only for those V with the same symmetries as U .P (V ) is determined with the maximum entropy principlegiven the above constraint.

No symmetry = Level repulsion

U

d

Pr(d)

d

Caltech, November 2004 – p.13

Page 50: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

Symmetries = Block diagonal

U = exp

{−iHth̄

}

=

U1

U2

. . .

Ud

U1 U2+ + . . . = U

d

d

Pr(d)

Mixture of i.i.d variables → Poisson distribution.

Caltech, November 2004 – p.14

Page 51: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

Symmetries = Block diagonal

U = exp

{−iHth̄

}

=

U1

U2

. . .

Ud

U1 U2+ + . . . = U

d

d

Pr(d)

Mixture of i.i.d variables → Poisson distribution.

Caltech, November 2004 – p.14

Page 52: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

With symmetries (regular): 1N Tr{U} = 1

N

√N = 1√

N

No symmetries (chaotic): 1N Tr{U} = 1

NO(1) = 1N

Requires accuracy 1√N

so overhead is N : square-root improvement

over brute force classical computation.

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.15

Page 53: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

With symmetries (regular): 1N Tr{U} = 1

N

√N = 1√

N

No symmetries (chaotic): 1N Tr{U} = 1

NO(1) = 1N

Requires accuracy 1√N

so overhead is N : square-root improvement

over brute force classical computation.

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.15

Page 54: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

With symmetries (regular): 1N Tr{U} = 1

N

√N = 1√

N

No symmetries (chaotic): 1N Tr{U} = 1

NO(1) = 1N

Requires accuracy 1√N

so overhead is N : square-root improvement

over brute force classical computation.

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.15

Page 55: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Looking for symmetries

1N Tr{U} = 1

N

j eiφj

= 1N

j ~vj

−30 −20 −10 0 10−30

−25

−20

−15

−10

−5

0

5

With symmetries (regular): 1N Tr{U} = 1

N

√N = 1√

N

No symmetries (chaotic): 1N Tr{U} = 1

NO(1) = 1N

Requires accuracy 1√N

so overhead is N : square-root improvement

over brute force classical computation.

D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003).

Caltech, November 2004 – p.15

Page 56: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Fidelity decay

Quantum map U , e.g. U = exp{−iHt}. How sensitive is U toperturbation?

Perturbed quantum map Up, e.g.

Up = U exp{−iδV }︸ ︷︷ ︸

K

Fidelity after n steps (Lodschmidt echo):

Fn(ψ) =∣∣〈ψ|(Un)†Un

p |ψ〉∣∣2

State independent signature (average over ψ):

Fn =

Fn(ψ)dψ =

∣∣Tr{(Un)†Un

p }∣∣2

+N

N2 +N

Caltech, November 2004 – p.16

Page 57: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Fidelity decay

Quantum map U , e.g. U = exp{−iHt}. How sensitive is U toperturbation?

Perturbed quantum map Up, e.g.

Up = U exp{−iδV }︸ ︷︷ ︸

K

Fidelity after n steps (Lodschmidt echo):

Fn(ψ) =∣∣〈ψ|(Un)†Un

p |ψ〉∣∣2

State independent signature (average over ψ):

Fn =

Fn(ψ)dψ =

∣∣Tr{(Un)†Un

p }∣∣2

+N

N2 +N

Caltech, November 2004 – p.16

Page 58: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Fidelity decay

Quantum map U , e.g. U = exp{−iHt}. How sensitive is U toperturbation?

Perturbed quantum map Up, e.g.

Up = U exp{−iδV }︸ ︷︷ ︸

K

Fidelity after n steps (Lodschmidt echo):

Fn(ψ) =∣∣〈ψ|(Un)†Un

p |ψ〉∣∣2

State independent signature (average over ψ):

Fn =

Fn(ψ)dψ =

∣∣Tr{(Un)†Un

p }∣∣2

+N

N2 +N

Caltech, November 2004 – p.16

Page 59: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Fidelity decay

Quantum map U , e.g. U = exp{−iHt}. How sensitive is U toperturbation?

Perturbed quantum map Up, e.g.

Up = U exp{−iδV }︸ ︷︷ ︸

K

Fidelity after n steps (Lodschmidt echo):

Fn(ψ) =∣∣〈ψ|(Un)†Un

p |ψ〉∣∣2

State independent signature (average over ψ):

Fn =

Fn(ψ)dψ =

∣∣Tr{(Un)†Un

p }∣∣2

+N

N2 +N

Caltech, November 2004 – p.16

Page 60: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Fidelity decay

Peres’ conjecture: Fn(ψ) is a signature of quantum chaos(dynamical instability).

Chaotic systems have an exponential decay exp(−Γn).Regular systems have a polynomial decay 1/poly(n).

Fidelity decay measures the relative randomness of U and K.Universal response to perturbation.“Simple” perturbation produce good signature of quantumchaos.

J. Emerson, Y.S. Weinstein, S. Lloyd, and D.G. Cory, 2002

Caltech, November 2004 – p.17

Page 61: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Fidelity decay

Peres’ conjecture: Fn(ψ) is a signature of quantum chaos(dynamical instability).

Chaotic systems have an exponential decay exp(−Γn).Regular systems have a polynomial decay 1/poly(n).

Fidelity decay measures the relative randomness of U and K.Universal response to perturbation.“Simple” perturbation produce good signature of quantumchaos.

J. Emerson, Y.S. Weinstein, S. Lloyd, and D.G. Cory, 2002

Caltech, November 2004 – p.17

Page 62: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Fidelity decay

Peres’ conjecture: Fn(ψ) is a signature of quantum chaos(dynamical instability).

Chaotic systems have an exponential decay exp(−Γn).Regular systems have a polynomial decay 1/poly(n).

Fidelity decay measures the relative randomness of U and K.Universal response to perturbation.“Simple” perturbation produce good signature of quantumchaos.

J. Emerson, Y.S. Weinstein, S. Lloyd, and D.G. Cory, 2002

Caltech, November 2004 – p.17

Page 63: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Trace circuit

D. Poulin, R. Blume-Kohout, R. Laflamme, and H. Ollivier, 2003.

s

U

H H 〈σk〉|0〉〈0|

1l

2n

k = y : 〈σy〉 =1

NIm [Tr{U}]

k = z : 〈σz〉 =1

NRe [Tr{U}]

Fn =

∣∣Tr{(Un)†Un

p }∣∣2

+N

N2 +N, Up = UK

All we have to do it replace U in the above circuit by (Un)†(UK)n.

Caltech, November 2004 – p.18

Page 64: Exploring Quantum Chaos with Quantum Information Processors · g iImf g iImf g 1 Ref g k = z : h˙zi = Trfˆ0 1˙zg = Ref g = Re[TrfˆUg] k = y : h˙yi = Trfˆ0 1˙yg = Imf g = Im[TrfˆUg]

Trace circuit

D. Poulin, R. Blume-Kohout, R. Laflamme, and H. Ollivier, 2003.

s

U

H H 〈σk〉|0〉〈0|

1l

2n

k = y : 〈σy〉 =1

NIm [Tr{U}]

k = z : 〈σz〉 =1

NRe [Tr{U}]

Fn =

∣∣Tr{(Un)†Un

p }∣∣2

+N

N2 +N, Up = UK

All we have to do it replace U in the above circuit by (Un)†(UK)n.

Caltech, November 2004 – p.18

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Quantum circuit

s s s s s s s s sH

U K U K U † U † U †KU

〈σk〉

︸ ︷︷ ︸

n

︸ ︷︷ ︸

n

|0〉1lN

... ...

s s sH

U K U KKU

〈σk〉

︸ ︷︷ ︸

n

|0〉1lN

...

Since the U ’s and U †’s annihilate each others when the K ’s areabsent.

Since the U † are made after the final measurement.

Measure Fn within accuracy ε with error probability p in time

nO

(log(1/p)

ε2

)

T

Caltech, November 2004 – p.19

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Quantum circuit

s s sH

U K U K U † U † U †KU

〈σk〉

︸ ︷︷ ︸

n

︸ ︷︷ ︸

n

|0〉1lN

... ...

Since the U ’s and U †’s annihilate each others when the K ’s areabsent.

s s sH

U K U KKU

〈σk〉

︸ ︷︷ ︸

n

|0〉1lN

...

Since the U ’s and U †’s annihilate each others when the K ’s areabsent.

Since the U † are made after the final measurement.

Measure Fn within accuracy ε with error probability p in time

nO

(log(1/p)

ε2

)

T

Caltech, November 2004 – p.19

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Quantum circuit

s s sH

U K U KKU

〈σk〉

︸ ︷︷ ︸

n

|0〉1lN

...

Since the U ’s and U †’s annihilate each others when the K ’s areabsent.

Since the U † are made after the final measurement.

Measure Fn within accuracy ε with error probability p in time

nO

(log(1/p)

ε2

)

T

Caltech, November 2004 – p.19

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Quantum circuit

s s sH

U K U KKU

〈σk〉

︸ ︷︷ ︸

n

|0〉1lN

...

Since the U ’s and U †’s annihilate each others when the K ’s areabsent.

Since the U † are made after the final measurement.

Measure Fn within accuracy ε with error probability p in time

nO

(log(1/p)

ε2

)

T

Caltech, November 2004 – p.19

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Experimental implementationColm Ryan, et al.(IQC)

Caltech, November 2004 – p.20

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Decoherence

We use the circuit to model a system interacting with anenvironment, the system is initially in a pure state α|0〉+ β|1〉:

s s s

U K U KKU ...

ρnρ0

1lN

ρ0 =

(

|α|2 αβ∗

α∗β |β|2

)

⇒ ρn =

(

|α|2 αβ∗γ(n)

α∗βγ(n)∗ |β|2

)

|γ(n)| ={Fn

}1/2: The decoherence rate is governed by the average

fidelity decay rate of the environment!

The decoherence rate depends on the strength of the coupling tothe environment but also on the dynamics of the environment.

Caltech, November 2004 – p.21

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Decoherence

We use the circuit to model a system interacting with anenvironment, the system is initially in a pure state α|0〉+ β|1〉:

s s s

U K U KKU ...

ρnρ0

1lN

ρ0 =

(

|α|2 αβ∗

α∗β |β|2

)

⇒ ρn =

(

|α|2 αβ∗γ(n)

α∗βγ(n)∗ |β|2

)

|γ(n)| ={Fn

}1/2: The decoherence rate is governed by the average

fidelity decay rate of the environment!

The decoherence rate depends on the strength of the coupling tothe environment but also on the dynamics of the environment.

Caltech, November 2004 – p.21

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Decoherence

We use the circuit to model a system interacting with anenvironment, the system is initially in a pure state α|0〉+ β|1〉:

s s s

U K U KKU ...

ρnρ0

1lN

ρ0 =

(

|α|2 αβ∗

α∗β |β|2

)

⇒ ρn =

(

|α|2 αβ∗γ(n)

α∗βγ(n)∗ |β|2

)

|γ(n)| ={Fn

}1/2: The decoherence rate is governed by the average

fidelity decay rate of the environment!

The decoherence rate depends on the strength of the coupling tothe environment but also on the dynamics of the environment.

Caltech, November 2004 – p.21

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Decoherence

We use the circuit to model a system interacting with anenvironment, the system is initially in a pure state α|0〉+ β|1〉:

s s s

U K U KKU ...

ρnρ0

1lN

ρ0 =

(

|α|2 αβ∗

α∗β |β|2

)

⇒ ρn =

(

|α|2 αβ∗γ(n)

α∗βγ(n)∗ |β|2

)

|γ(n)| ={Fn

}1/2: The decoherence rate is governed by the average

fidelity decay rate of the environment!

The decoherence rate depends on the strength of the coupling tothe environment but also on the dynamics of the environment.

Caltech, November 2004 – p.21

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Entanglement

s s s

U2 K2 Uk KkK1U1...

α|0〉+ β|1〉

1lN

Using the "phase kick-back", we see that the final state is

ρk =1

N

j

|αj〉〈αj | ⊗ |φj〉〈φj |

where |αj〉 = α|0〉+ βeisj |1〉 and the eisj are the eigenvalues ofS = UkPk . . . U2P2U1P1U

†1U

†2 . . . U

†k .

This circuit does not produce any entanglement between the probequbit and the rest, throughout the computation

Caltech, November 2004 – p.22

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Entanglement

s s s

U2 K2 Uk KkK1U1...

α|0〉+ β|1〉

1lN

Using the "phase kick-back", we see that the final state is

ρk =1

N

j

|αj〉〈αj | ⊗ |φj〉〈φj |

where |αj〉 = α|0〉+ βeisj |1〉 and the eisj are the eigenvalues ofS = UkPk . . . U2P2U1P1U

†1U

†2 . . . U

†k .

This circuit does not produce any entanglement between the probequbit and the rest, throughout the computation

Caltech, November 2004 – p.22

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Entanglement

s s s

U2 K2 Uk KkK1U1...

α|0〉+ β|1〉

1lN

Using the "phase kick-back", we see that the final state is

ρk =1

N

j

|αj〉〈αj | ⊗ |φj〉〈φj |

where |αj〉 = α|0〉+ βeisj |1〉 and the eisj are the eigenvalues ofS = UkPk . . . U2P2U1P1U

†1U

†2 . . . U

†k .

This circuit does not produce any entanglement between the probequbit and the rest, throughout the computation

Caltech, November 2004 – p.22

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Entanglement

Decoherence does not require entanglement between thesystem and its environment: classical correlations are enough.

Is entanglement necessary for quantum computationalspeed-up? (If speed-up there is.)

Pure states: YES.Mixed states: ?This is an argument in favor of no... but not a proof.

Is there entanglement between other parts of the informationprocessor? (Depends on the system of interest, i.e. U .)

If no, is there a classical dynamics which can provide thesequence of "classical" states?

Caltech, November 2004 – p.23

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Entanglement

Decoherence does not require entanglement between thesystem and its environment: classical correlations are enough.

Is entanglement necessary for quantum computationalspeed-up? (If speed-up there is.)

Pure states: YES.Mixed states: ?This is an argument in favor of no... but not a proof.

Is there entanglement between other parts of the informationprocessor? (Depends on the system of interest, i.e. U .)

If no, is there a classical dynamics which can provide thesequence of "classical" states?

Caltech, November 2004 – p.23

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Entanglement

Decoherence does not require entanglement between thesystem and its environment: classical correlations are enough.

Is entanglement necessary for quantum computationalspeed-up? (If speed-up there is.)

Pure states: YES.Mixed states: ?This is an argument in favor of no... but not a proof.

Is there entanglement between other parts of the informationprocessor? (Depends on the system of interest, i.e. U .)

If no, is there a classical dynamics which can provide thesequence of "classical" states?

Caltech, November 2004 – p.23

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Entanglement

Decoherence does not require entanglement between thesystem and its environment: classical correlations are enough.

Is entanglement necessary for quantum computationalspeed-up? (If speed-up there is.)

Pure states: YES.Mixed states: ?This is an argument in favor of no... but not a proof.

Is there entanglement between other parts of the informationprocessor? (Depends on the system of interest, i.e. U .)

If no, is there a classical dynamics which can provide thesequence of "classical" states?

Caltech, November 2004 – p.23

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Conclusion

DQC1 is interesting.Measure the form factors to learn about symmetries.Measure fidelity decay to learn about dynamical stability.

Experimental benchmarking of quantum informationprocessing.

Decoherence rate is influenced by the dynamical properties ofthe environment.

Decoherence does not require entenglement.

Entenglement is not the key ingredient to quantumcomputational speed-up.

We can learn about fundamental physics by working onquantum computation, and vice versa.

Caltech, November 2004 – p.24

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References

David Poulin, Raymond Laflamme, G.J. Milburn, and JuanPablo Paz, “Testing integrability with a single bit of quantuminformation”, Phys. Rev. A 68, 22302 (2003).

David Poulin, Robin Blume-Kohout, Raymond Laflamme, andHarold Ollivier, “Exponential speed-up with a single bit ofquantum information: Measuring the average fidelity decay”,Phys. Rev. Lett. 92 177906 (2004).

Joseph Emerson, Seth Lloyd, David Poulin, and David Cory,“Estimation of the Local Density of States on a QuantumComputer”, Phys. Rev. A 69 50305 (2004).

Caltech, November 2004 – p.25