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

Exploring Quantum Chaos withQuantum Information Processors

David Poulin

Institute for Quantum Computing

Perimeter Institute for Theoretical Physics

Caltech, November 2004 – p.1

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.

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.

Readout problem

U(t) ≈ Un . . . U2U1︸︷︷︸

Universal set

Readout problem

U(t) ≈ Un . . . U2U1︸︷︷︸

Universal set

Readout problem

U(t) ≈ Un . . . U2U1︸︷︷︸

Universal set

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

Readout problem

U(t) ≈ Un . . . U2U1︸︷︷︸

Universal set

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

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

1l− β∑

ωiσzi

Apply an inhomogeneous magnetic field to induce a randomphase:

ρ→ 1

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.

−βH .

H '∑

i ~µi · ~B +∑

ρ ' 1

1l− β∑

ωiσzi

ρ→ 1

1l +βωn

2n|0 . . . 00〉〈0 . . . 00|

−βH .

H '∑

i ~µi · ~B +∑

ρ ' 1

1l− β∑

ωiσzi

ρ→ 1

1l +βωn

2n|0 . . . 00〉〈0 . . . 00|

−βH .

H '∑

i ~µi · ~B +∑

ρ ' 1

1l− β∑

ωiσzi

ρ→ 1

1l +βωn

2n|0 . . . 00〉〈0 . . . 00|

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.

(1− ε

21l + ε|0〉〈0|

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

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

Add restrictions to the “standard” model of quantumcomputation

1. Only a single pseudo-pure qubit.

(1− ε

21l + ε|0〉〈0|

(1− ε

21l + ε|0〉〈0|

(1− ε

21l + ε|0〉〈0|

(1− ε

21l + ε|0〉〈0|

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|︸︷︷︸

Remarks:

|00 . . . 0〉〈00 . . . 0|︸︷︷︸

Remarks:

|00 . . . 0〉〈00 . . . 0|︸︷︷︸

Remarks:

|00 . . . 0〉〈00 . . . 0|︸︷︷︸

Scattering circuit

H H 〈σk〉|0〉〈0|

ργ = Tr{Uρ}

(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.

Scattering circuit

H H 〈σk〉|0〉〈0|

ργ = Tr{Uρ}

k = z : 〈σz〉 = Tr{ρ′

k = y : 〈σy〉 = Tr{ρ′

Scattering circuit

H H 〈σk〉|0〉〈0|

ργ = Tr{Uρ}

k = z : 〈σz〉 = Tr{ρ′

k = y : 〈σy〉 = Tr{ρ′

Scattering circuit

H H 〈σk〉|0〉〈0|

ργ = Tr{Uρ}

k = z : 〈σz〉 = Tr{ρ′

k = y : 〈σy〉 = Tr{ρ′

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.

NTr{U} =

Scattering circuit

NTr{U} =

Scattering circuit

NTr{U} =

Scattering circuit

NTr{U} =

Scattering circuit

NTr{U} =

Scattering circuit

NTr{U} =

Doing the obvious thing

1N Tr{U} = 1

j eiφj

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

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).

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

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).

Quantum control

Looking for symmetries

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

Looking for symmetriesList of resonances

Level spacing

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

f(U) '∫f(V )P (V )dV .

Symmetries = Block diagonal

U = exp

{−iHth̄

U1 U2+ + . . . = U

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

Symmetries = Block diagonal

U = exp

{−iHth̄

U1 U2+ + . . . = U

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

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

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

√N = 1√

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.

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

√N = 1√

NO(1) = 1N

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

√N = 1√

NO(1) = 1N

1N Tr{U} = 1

j eiφj

−30 −20 −10 0 10−30

√N = 1√

NO(1) = 1N

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 }︸︷︷︸

Fidelity after n steps (Lodschmidt echo):

Fn(ψ) =∣∣〈ψ|(Un)†Un

p |ψ〉∣∣2

State independent signature (average over ψ):

Fn(ψ)dψ =

∣∣Tr{(Un)†Un

p }∣∣2

Fidelity decay

p |ψ〉∣∣2

Fn(ψ)dψ =

∣∣Tr{(Un)†Un

p }∣∣2

Fidelity decay

p |ψ〉∣∣2

Fn(ψ)dψ =

∣∣Tr{(Un)†Un

p }∣∣2

Fidelity decay

p |ψ〉∣∣2

Fn(ψ)dψ =

∣∣Tr{(Un)†Un

p }∣∣2

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

Fidelity decay

Trace circuit

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

H H 〈σk〉|0〉〈0|

k = y : 〈σy〉 =1

NIm [Tr{U}]

k = z : 〈σz〉 =1

NRe [Tr{U}]

∣∣Tr{(Un)†Un

p }∣∣2

N2 +N, Up = UK

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

Trace circuit

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

H H 〈σk〉|0〉〈0|

k = y : 〈σy〉 =1

NIm [Tr{U}]

k = z : 〈σz〉 =1

NRe [Tr{U}]

∣∣Tr{(Un)†Un

p }∣∣2

N2 +N, Up = UK

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

Quantum circuit

s s s s s s s s sH

U K U K U † U † U †KU

〈σk〉

︸︷︷︸

|0〉1lN

... ...

s s sH

U K U KKU

〈σk〉

︸︷︷︸

|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

(log(1/p)

Quantum circuit

s s sH

U K U K U † U † U †KU

〈σk〉

︸︷︷︸

|0〉1lN

... ...

s s sH

U K U KKU

〈σk〉

︸︷︷︸

|0〉1lN

(log(1/p)

Quantum circuit

s s sH

U K U KKU

〈σk〉

︸︷︷︸

|0〉1lN

(log(1/p)

Quantum circuit

s s sH

U K U KKU

〈σk〉

︸︷︷︸

|0〉1lN

(log(1/p)

Experimental implementationColm Ryan, et al.(IQC)

Decoherence

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

U K U KKU ...

ρnρ0

|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.

Decoherence

U K U KKU ...

ρnρ0

|α|2 αβ∗

α∗β |β|2

⇒ ρn =

|α|2 αβ∗γ(n)

α∗βγ(n)∗ |β|2

|γ(n)| ={Fn

Decoherence

U K U KKU ...

ρnρ0

|α|2 αβ∗

α∗β |β|2

⇒ ρn =

|α|2 αβ∗γ(n)

α∗βγ(n)∗ |β|2

|γ(n)| ={Fn

Decoherence

U K U KKU ...

ρnρ0

|α|2 αβ∗

α∗β |β|2

⇒ ρn =

|α|2 αβ∗γ(n)

α∗βγ(n)∗ |β|2

|γ(n)| ={Fn

Entanglement

U2 K2 Uk KkK1U1...

α|0〉+ β|1〉

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

ρk =1

|αj〉〈αj | ⊗ |φj〉〈φj |

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

†2 . . . U

†k .

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

Entanglement

U2 K2 Uk KkK1U1...

α|0〉+ β|1〉

ρk =1

|αj〉〈αj | ⊗ |φj〉〈φj |

†2 . . . U

†k .

Entanglement

U2 K2 Uk KkK1U1...

α|0〉+ β|1〉

ρk =1

|αj〉〈αj | ⊗ |φj〉〈φj |

†2 . . . U

†k .

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?

Entanglement

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.

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).

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

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