quantum’speedup’by’adiabac’state’ transformaons’and ...€¦ ·...

Quantum speedup by adiaba/c state transforma/ons and quantum annealing

Rolando D. Somma Los Alamos Na/onal Laboratory [email protected]

Joint work with Sergio Boixo Maria Kieferova Emanuel Knill Daniel Nagaj Guanglei Xu HT Chiang

References: S Boixo, E Knill, RDS, QIC 9, 0833 (2009) S Boixo, E Knill, RDS, arXiv: 1005.3034 (2010) RDS, S Boixo, arXiv: 1110.2494 (2011) -‐ SICOMP RDS, D Nagaj, M Kieferova, PRL 109, 050501 (2012) Chiang, Xu, RDS, PRA 89, 012314 (2014)

Quantum Algorithms in Hamiltonian-‐based Models of QC

0

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ψ(T)

!!

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ψ(0) = 00...0 !is!initial!stateH(t)!is!the!time - dependent!Hamiltonian!path

!!

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initial!state : !!ψ(0) = 00...0

unitary!evolution : !!!!!!U = τ exp −i H(t)dt0

T

∫&

' (

)

* +

, - .

/ 0 1

final!state : !!!!ψ(T) = a00...0 00...0 + a10...0 10...0 + ...+ a11...1 11...1

measurement : !!! σ !, !!Pr(σ) =| aσ |2

Coherent evolu/on

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i∂t ψ(t) = H(t)ψ(t)

0

0

0

0

/me

Adiaba/c Quantum Compu/ng

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H0

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Hf

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H(t)

!!

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TAQC >>1

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H0

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Hf

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H(t)

€

t


€

H0

€

Hf

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H(t)

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H0

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Hf

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H(t)

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tInput Hamiltonian

!!

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TAQC >>1


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H0

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Hf

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H(t)

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H0

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Hf

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H(t)

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tInput Hamiltonian

!!

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Eigenvalues : !!!λ0(t) < λ1(t) ≤ λ2(t)...Ground!state : !! φ(t)

!!

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TAQC >>1


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H0

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Hf

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H(t)

Adiaba1c theorem (Born-‐Fock) A quantum system ini/ally in its ground state will remain in its instantaneous ground state in the limit in which the perturba/on rate vanishes (T >> 1).

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H0

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Hf

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H(t)

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tInput Hamiltonian

!!

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Eigenvalues : !!!λ0(t) < λ1(t) ≤ λ2(t)...Ground!state : !! φ(t)

!!

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TAQC >>1

Quantum Adiaba/c Approxima/on

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H0 = H(0)

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H(1) = Hf

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H(s(t)) = H(t)φ(s(t)) = φ(t)

Standard quantum adiaba1c approxima1on [1]

!!

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TAQC ∝maxs∂s2H(s)Δ(s)( )2

,∂sH(s)( )

2

Δ(s)( )3

%

& '

( '

)

* '

+ '

!!

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ψ(TAQC) ≈ε φ(1)

0 1

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λ0(s)

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λ1(s)

!!

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Eigenvalues!of!H(s)

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φ(s)

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Δ(s)€

s

!!

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We!rescale!the!time!so!that!s = t/TAQC , !0 ≤ s ≤1

[1] S. Jansen, M. Ruskai, and R. Seiler, J. Math. Phys.48, 102111(2007). +++

Adiaba/c Quantum Compu/ng: Equivalence and complexity Any quantum circuit can be simulated with an adiaba/c evolu/on [2]:

...

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U1

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U2

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UL

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φ0

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φL

!!

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H(1) = f (U1,...,UL )!; ! φL ψ(1) ≈1Size of the system is polynomial in L Simple one or two-‐qubit gates

[2] D. Aharonov, et.al., SICOMP 37, 166 (2007). A. Mizel, D. Lidar, M. Mitchell, PRL 99, 070502 (2007).


...

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U1

€

U2

€

UL

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φ0

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φL

!!

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H(1) = f (U1,...,UL )!; ! φL ψ(1) ≈1Size of the system is polynomial in L

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U(T) =ℑ[exp(−i H(t)dt0

T

∫ )]

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U(T) ≈U1...UL

Simple one or two-‐qubit gates

How many elementary gates do we need to simulate the evolu/on operator?



...

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U1

€

U2

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UL

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φ0

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φL

!!

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H(1) = f (U1,...,UL )!; ! φL ψ(1) ≈1Size of the system is polynomial in L

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U(T) =ℑ[exp(−i H(t)dt0

T

∫ )]

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U(T) ≈U1...UL

Simple one or two-‐qubit gates

How many elementary gates do we need to simulate the evolu/on operator?

•  Sparse •  Bounded norm

!!

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L! = !poly(T)


Adiaba/c Quantum Compu/ng: Why?

Why do we really care about AQC?

Adiaba/c Quantum Compu/ng: Why


Inherent robustness? Fault tolerance?




Natural model for solving certain problems, such as combinatorial op/miza/on




Natural model for solving certain problems, such as combinatorial op/miza/on

Provable quantum speedups?

Faster ways to prepare the final ground state?

!!

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evolution!time!T << TAQC!???

Quantum Adiaba/c Approxima/on: Problems Issues with standard adiaba/c approxima/ons: They may result in undesirably large and unnecessary costs!

Quantum Adiaba/c Approxima/on: Problems Issues with standard adiaba/c approxima/ons: They may result in undesirably large and unnecessary costs!

In the last few years, we developed several methods to prepare the final ground state in /me much less than that given by AQC. Our methods resulted in provable quantum speedups of several classical algorithms, such as Monte Carlo.

Quantum Adiaba/c Approxima/on: New methods Issues with standard adiaba/c approxima/ons: They may result in undesirably large and unnecessary costs!


•  Method 1: Evolu/on randomiza/on (Boixo, Knill, Xu)

•  Method 2: Measurement based (Boixo, Knill)

•  Method 3: Diaba/c transi/ons (Nagaj, Kieferova)

Quantum Adiaba/c Approxima/on: New methods Issues with standard adiaba/c approxima/ons: They may result in undesirably large and unnecessary costs!


•  Method 1: Evolu/on randomiza/on (Boixo, Knill, Xu)

•  Method 2: Measurement based (Boixo, Knill)

•  Method 3: Diaba/c transi/ons (Nagaj, Kieferova)

“adiaba/c state transforma/ons”

“quantum annealing”

As in AQC, the goal is to prepare the final ground state (or any eigenstate) from the ini/al one by sequen/ally preparing the ground states (eigenstates) along the path.

1. Evolu/on randomiza/on [3]

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φ(0) = φ0€

φ1

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φ2

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φm = φ(1)

Assume a discre/za/on for the eigenpath (path of ground states)

…

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δ

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φ j = φ(s j )€

{0 = s0,s1,...,sm =1}

[3] S Boixo, E Knill, RDS, QIC 9, 0833 (2009). Chiang, Xu, RDS, PRA 89, 012314 (2014)

1. Evolu/on randomiza/on: “Measurements”

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φ(0) = φ0€

φ1

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φ2

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φm = φ(1)

Discre/za/on for the eigenpath

A sequence of projec/ve measurements into the ground states prepares the final ground state with high probability

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δ

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φ j = φ(s j ) €

{0 = s0,s1,...,sm =1}

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φ j−1 φ j ≥1−δ 2


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φ(0) = φ0€

φ1✕

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Pr φ1[ ] ≥1−δ 2

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δ

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φ j = φ(s j )


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φ(0) = φ0€

φ1

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φ2✕

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Pr φ2[ ] ≥1− 2δ 2

✕

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δ

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φ j = φ(s j )

1. Evolu/on randomiza/on: Path length

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φ(0) = φ0€

φ1

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φ2

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φm = φ(1)

✕ ✕ ✕ ✕

✕ ✕

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Pr φm[ ] ≥1−mδ 2 =1− Lδ

It suffices to pick δ=ε/L for overall error ε

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δ

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φ j = φ(s j )

L=m.δ

Path length

1. Evolu/on randomiza/on: Cost

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φ(0) = φ0€

φ1

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φ2

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φm = φ(1)

✕ ✕ ✕ ✕

✕ ✕

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Pr φm[ ] ≥1−mδ 2 =1− Lδ

It suffices to pick δ=ε/L for overall error ε

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δ

!!

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m =L2

ε!; !!m =

ds ∂sφ(s) ∂sφ(s)0

1

∫ε

Number of projec/ve measurements:

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φ j = φ(s j )

€

ε

Simula/on of projec/ve measurement by evolu/on randomiza/on

1: Evolu/on randomiza/on: Simula/on of measurements

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e−iH j t φ j φ j e+iH j t = φ j φ j

e−iH j t φ j φ j⊥ e+iH j t = eiΔ' t φ j φ j

⊥


Orthogonal eigenstate

Δ ' ≥ Δ

€



⊥




Evolu/on randomiza/on can eliminate coherences, thus simula/ng a projec/ve measurement

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global!cost : !!!Trand = τ.m ∝L2

εΔ

Δ ' ≥ Δ

!!

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∃!!f (t)!!such!that

dt. f (t).0

τ

∫ e−iH j t φ j φ j⊥ e+ iH j t = 0

τ ≈1/Δ suffices from Fourier analysis

€



⊥

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τ ≈1/Δ

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f (t)



1. Evolu/on randomiza/on: Total cost


Evolu/on randomiza/on can eliminate coherences, thus simula/ng a projec/ve measurement

!!

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global!cost : !!!Trand = τ.m ∝L2

εΔ

Δ ' ≥ Δ

!!

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∃!!f (t)!!such!that

dt. f (t).0

τ

∫ e−iH j t φ j φ j⊥ e+ iH j t = 0

τ ≈1/Δ suffices from Fourier analysis

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⊥

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τ ≈1/Δ

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f (t)

!!

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Trand << TAQC!in!many!examples

[3] S Boixo, E Knill, RDS, QIC 9, 0833 (2009). Chiang, Xu, RDS, PRA 89, 012314 (2014)

2. Measurement based

The main different with the other method is that we will simulate the projec/ve measurements in the ground states in a different way. Rather than needing δ=ε/L, we will be able to choose a constant δ and reduce the cost (number of points in the discre/za/on).

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φ(0) = φ0€

φ1

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φ2

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φm = φ(1)

…

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δ

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φ j = φ(s j )

2. Measurement based: Basic steps

One-‐step state transforma1ons

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Goal : !prepare! φ j !!from! φ j−1 !using!reflection!

!!!!!!!!!!!oracles!R j−1 =1− 2 φ j−1 φ j−1 !and!Rj =1− 2 φ j φ j



!!

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Pseudocode:

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1.!Perform!a!projective!measurement!of! φ j !

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φ j−1



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φ j

!!

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Pseudocode:

!!

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φ j−1

!!

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2.!If!successful : !STOP



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φ j

!!

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Pseudocode:

!!

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!!

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φ j⊥

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φ j−1

!!

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3.!Else : !Apply!R j ,1



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φ j

!!

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Pseudocode:

!!

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!!

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!!

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4.!Go!to!1.€

φ j⊥

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φ j−1

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" ψ

!!

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3.!Else : !Apply!R j ,1


H H

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R j−1

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R j€

0

!!

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If!0!STOPIf!1!Execute!again

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ψ

!!

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p0 = φ j φ j−1

2≥1−δ 2!!; !!p = & ψ φ j

2!!; !!q =1− p

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!!

€



€

φ j−1

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φ j−1

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φ j

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ψ'


H H

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R j−1

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R j€

0

!!

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If!0!STOPIf!1!Execute!again

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ψ

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!!

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If!!p0 >1/3!⇒ ! n !!is!order!1!and!all!moments!are!bounded!!


!!

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φ j−1

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φ j−1

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φ j

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ψ'

!!

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p0 = φ j φ j−1

2≥1−δ 2!!; !!p = & ψ φ j

2!!; !!q =1− p

2. Measurement based: How to implement reflec/ons

Assume we know the ground state energy

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H j φ j = 0

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R j =1− 2 φ j φ j

Quantum Phase Es/ma/on Algorithm

Quantum Phase Es/ma/on Algorithm (INVERSE)

Ancilliary qubits

Control on

|00…0>

FLIP SIGN

The phase es/ma/on algorithm needs to resolve eigenvalues above the gap. Thus, it requires a cost of 1/Δ:

!!

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e−iH j t!! !; !!t ≈1/Δ

2. Measurement based: Cost

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global!cost : !!!TMB ∝L.log(L /ε)

ΔAlmost op/mal!

2. Measurement based: Cost

!!

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global!cost : !!!TMB ∝L.log(L /ε)

ΔAlmost op/mal!

!!

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T ≥ L /Δ!![4]

!!

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TMB << Trand << TAQC

[4] S. Boixo and R. Somma, PRA 81, 032308 (2010).

3. Diaba/c transi/ons: Glued-‐Trees problem

The (randomly) Glued-‐Trees Problem (GT) [5]

Each vertex is randomly labeled as a(VERTEX)

j=0 j=2n+1

j=1

j=n j=n+1 # of ver/ces: 2n+2 -‐ 2

[5] A. M. Childs, R. Cleve, et.al., in Proc. 35th Annual ACM Symp. Theo. of Comp., p. 59 (2003)

(given) (unknown)

(given oracle access to A)




j=0 j=2n+1

j=1

j=n j=n+1 # of ver/ces: 2n+2 -‐ 2

(given) (unknown)

Classically it takes /me exponen/al in n to find the exit. Quantumly it can be done in /me polynomial in n






j=0 j=2n+1

j=1

j=n j=n+1 # of ver/ces: 2n+2 -‐ 2

(given)


(unknown)

Can we solve this problem adiaba/cally?


3. Diaba/c transi/ons: Glued-‐Trees problem Exponen/al speedups [6]

[6] “Quantum speedup by quantum annealing”, RS, Nagaj, Kieferova, PRL 109, 050501 (2012)

The (randomly) Glued-‐Trees Problem (GT)


j=0 j=2n+1

j=1

j=n j=n+1 # of ver/ces: 2n+2 -‐ 2

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H0

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Hf

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H(t)

!!

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H(s) = (1− s) ENTRANCE ENTRANCE + s(1− s)A + s EXIT EXIT

!!

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λ0(s)€

λ1(s)

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λ2(s)

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λ3(s)

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Δ 01 ≈ exp(−n)

Entrance Exit

3. Diaba/c transi/ons: Quantum annealing

!!

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λ0(s)€

λ1(s)

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λ2(s)

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λ3(s)

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Δ12 ≈1/ poly(n)Entrance Exit

3. Diaba/c transi/ons: Quantum annealing [6]



!!

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Adiaba/cally: exponen/al /me

Entrance Exit



!!

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Entrance Exit

Adiaba/c Adiaba/c Adiaba/c

Fast! Fast!



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Diaba1c transi1ons: polynomial 1me !!

Provable quantum speedup

Entrance Exit


3. Diaba/c transi/ons: Importance & generaliza/ons

Why is the result important? (Addi/onal reasons) •  Recently, mo/vated by our results, a similar property on the spectrum of other Hamiltonians for solving MAX 2 SAT was shown [7]. This provided faster annealing algorithms for this problem.

[7] E.Crosson, E. Farhi, C. Lin, H. Lin, P. Shor, arXiv: 1401.7320 (2014)

Non-‐adiaba/c regime “hard” instance 20 bits

3. Diaba/c transi/ons: Importance & generaliza/ons

Why is the result important? (Addi/onal reasons) •  The Hamiltonians involved do not suffer from the so-‐called sign problem. Then, classical techniques such as quantum Monte Carlo can be used in these cases. Can quantum annealing outperform quantum Monte Carlo? When the gaps are big, this ques/on remains unanswered.

Conclusions

•  We presented methods for adiaba/c state transforma/ons that, in some cases, are (almost) op/mal and achieve a cost of L/Δ. Can we always achieve such cost?

•  We presented a method to avoid the overheads due to extremely small gaps in adiaba/c state transforma/ons. This method aims at adiaba/cally decoupling subspaces and uses diaba/c transi/ons to excited states.

•  The techniques we presented allow for provable polynomial and exponen/al speedups.

quantum’speedup’by’adiabac’state’ transformaons’and ...€¦ ·...

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