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Quantum Thermodynamics on a QuantumComputer
David Poulin
Département de PhysiqueUniversité de Sherbrooke
American Physical Society’s March MeetingPittsburgh, March 2008
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 1 / 20
Motivation
Outline
1 Motivation
2 Existing methods for classical ground state
3 The obvious method
4 How to fix it...
5 Future work
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 2 / 20
Motivation
Quantum information meets many-particle physics
Topologically ordered systems and long-lived quantum memories.Quantum error correction and statistical mechanics.Complexity of determining ground state energy.Entropy area laws for gapped systems.Phase transition in adiabatic quantum computation.Entanglement renormalisation and related approches fornumerical simulations.Quantum circuits and Ising partition functions.Coherent many-body devices for qubits.etc.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 3 / 20
Motivation
Quantum simulation
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 4 / 20
Motivation
Quantum simulation
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 4 / 20
Motivation
Quantum simulation
Universal quantum computer
Collection of n 2-level quantum systems, H = (C2)⊗n.
Set of one- and two-body gates, e.g. ei π8 σ(k)z ,ei π2 σ
(k)x ,ei π2 σ
(k)z σ
(k+1)z .
For a system of n interacting quantum particles, e.g.H =
∑〈i,j〉 h
(i,j)
The unitary transformation U = e−iHt can be approximated withinaccuracy ε by an efficient quantum circuit:
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 5 / 20
Motivation
Quantum simulation
Universal quantum computer
Collection of n 2-level quantum systems, H = (C2)⊗n.
Set of one- and two-body gates, e.g. ei π8 σ(k)z ,ei π2 σ
(k)x ,ei π2 σ
(k)z σ
(k+1)z .
For a system of n interacting quantum particles, e.g.H =
∑〈i,j〉 h
(i,j)
The unitary transformation U = e−iHt can be approximated withinaccuracy ε by an efficient quantum circuit:
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 5 / 20
Motivation
Quantum simulation
Universal quantum computer
Collection of n 2-level quantum systems, H = (C2)⊗n.
Set of one- and two-body gates, e.g. ei π8 σ(k)z ,ei π2 σ
(k)x ,ei π2 σ
(k)z σ
(k+1)z .
For a system of n interacting quantum particles, e.g.H =
∑〈i,j〉 h
(i,j)
The unitary transformation U = e−iHt can be approximated withinaccuracy ε by an efficient quantum circuit:
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 5 / 20
Motivation
Quantum simulation
Universal quantum computer
Collection of n 2-level quantum systems, H = (C2)⊗n.
Set of one- and two-body gates, e.g. ei π8 σ(k)z ,ei π2 σ
(k)x ,ei π2 σ
(k)z σ
(k+1)z .
For a system of n interacting quantum particles, e.g.H =
∑〈i,j〉 h
(i,j)
The unitary transformation U = e−iHt can be approximated withinaccuracy ε by an efficient quantum circuit:
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 5 / 20
Motivation
Quantum simulation
Universal quantum computer
Collection of n 2-level quantum systems, H = (C2)⊗n.
Set of one- and two-body gates, e.g. ei π8 σ(k)z ,ei π2 σ
(k)x ,ei π2 σ
(k)z σ
(k+1)z .
For a system of n interacting quantum particles, e.g.H =
∑〈i,j〉 h
(i,j)
The unitary transformation U = e−iHt can be approximated withinaccuracy ε by an efficient quantum circuit:
e = -iHt U
A
# of gates = poly(n,t,ε )-1
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 5 / 20
Motivation
A notoriously hard problem
A quantum computer is thus a powerful ’integration tool’ — itsolves Schrödinger’s equation ψ(t) = Uψ(0).This integration tool is not of much use if we cannot specify theinitial conditions; what is ψ(0)?Physically relevant are the low energy states of H.
Preparing such states is a difficult problem:
QuestionIs the lowest eigenvalue of H less than α or greater than β whereβ − α > 1/poly(n).
Classical case, i.e. [h(ij),h(ik)] = 0, is NP-complete (Barahona).Remains hard even when β − α ∝ n (PCP).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 6 / 20
Motivation
A notoriously hard problem
A quantum computer is thus a powerful ’integration tool’ — itsolves Schrödinger’s equation ψ(t) = Uψ(0).This integration tool is not of much use if we cannot specify theinitial conditions; what is ψ(0)?Physically relevant are the low energy states of H.
Preparing such states is a difficult problem:
QuestionIs the lowest eigenvalue of H less than α or greater than β whereβ − α > 1/poly(n).
Classical case, i.e. [h(ij),h(ik)] = 0, is NP-complete (Barahona).Remains hard even when β − α ∝ n (PCP).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 6 / 20
Motivation
A notoriously hard problem
A quantum computer is thus a powerful ’integration tool’ — itsolves Schrödinger’s equation ψ(t) = Uψ(0).This integration tool is not of much use if we cannot specify theinitial conditions; what is ψ(0)?Physically relevant are the low energy states of H.
Preparing such states is a difficult problem:
QuestionIs the lowest eigenvalue of H less than α or greater than β whereβ − α > 1/poly(n).
Classical case, i.e. [h(ij),h(ik)] = 0, is NP-complete (Barahona).Remains hard even when β − α ∝ n (PCP).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 6 / 20
Motivation
A notoriously hard problem
A quantum computer is thus a powerful ’integration tool’ — itsolves Schrödinger’s equation ψ(t) = Uψ(0).This integration tool is not of much use if we cannot specify theinitial conditions; what is ψ(0)?Physically relevant are the low energy states of H.
Preparing such states is a difficult problem:
QuestionIs the lowest eigenvalue of H less than α or greater than β whereβ − α > 1/poly(n).
Classical case, i.e. [h(ij),h(ik)] = 0, is NP-complete (Barahona).Remains hard even when β − α ∝ n (PCP).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 6 / 20
Motivation
A notoriously hard problem
A quantum computer is thus a powerful ’integration tool’ — itsolves Schrödinger’s equation ψ(t) = Uψ(0).This integration tool is not of much use if we cannot specify theinitial conditions; what is ψ(0)?Physically relevant are the low energy states of H.
Preparing such states is a difficult problem:
QuestionIs the lowest eigenvalue of H less than α or greater than β whereβ − α > 1/poly(n).
Classical case, i.e. [h(ij),h(ik)] = 0, is NP-complete (Barahona).Remains hard even when β − α ∝ n (PCP).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 6 / 20
Motivation
A notoriously hard problem
A quantum computer is thus a powerful ’integration tool’ — itsolves Schrödinger’s equation ψ(t) = Uψ(0).This integration tool is not of much use if we cannot specify theinitial conditions; what is ψ(0)?Physically relevant are the low energy states of H.
Preparing such states is a difficult problem:
QuestionIs the lowest eigenvalue of H less than α or greater than β whereβ − α > 1/poly(n).
Classical case, i.e. [h(ij),h(ik)] = 0, is NP-complete (Barahona).Remains hard even when β − α ∝ n (PCP).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 6 / 20
Existing methods for classical ground state
Outline
1 Motivation
2 Existing methods for classical ground state
3 The obvious method
4 How to fix it...
5 Future work
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 7 / 20
Existing methods for classical ground state
Simulated Annealing
Imitates an initially hot metal (random state) that is slowly cooleddown. For 0 = β0 ≤ β1 ≤ . . . ≤ β` ' n,
12n →
e−β1Ei
Z(β1)→ . . .→ e−β`Ei
Z(β`)
If cooling is too fast, the system can become trapped in a localminimum. Running time depends on the energy landscape.In the worst case, the running time is proportional to the numberof states D = 2n.A quantum simulated annealing algorithm offers a
√· speed-upover all classical annealing processes. (Somma et al.)Used for all sort of (classical) combinatorial optimization problems.These techniques are only suited for classical problems.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 8 / 20
Existing methods for classical ground state
Simulated Annealing
Imitates an initially hot metal (random state) that is slowly cooleddown. For 0 = β0 ≤ β1 ≤ . . . ≤ β` ' n,
12n →
e−β1Ei
Z(β1)→ . . .→ e−β`Ei
Z(β`)
If cooling is too fast, the system can become trapped in a localminimum. Running time depends on the energy landscape.In the worst case, the running time is proportional to the numberof states D = 2n.A quantum simulated annealing algorithm offers a
√· speed-upover all classical annealing processes. (Somma et al.)Used for all sort of (classical) combinatorial optimization problems.These techniques are only suited for classical problems.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 8 / 20
Existing methods for classical ground state
Simulated Annealing
Imitates an initially hot metal (random state) that is slowly cooleddown. For 0 = β0 ≤ β1 ≤ . . . ≤ β` ' n,
12n →
e−β1Ei
Z(β1)→ . . .→ e−β`Ei
Z(β`)
If cooling is too fast, the system can become trapped in a localminimum. Running time depends on the energy landscape.In the worst case, the running time is proportional to the numberof states D = 2n.A quantum simulated annealing algorithm offers a
√· speed-upover all classical annealing processes. (Somma et al.)Used for all sort of (classical) combinatorial optimization problems.These techniques are only suited for classical problems.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 8 / 20
Existing methods for classical ground state
Simulated Annealing
Imitates an initially hot metal (random state) that is slowly cooleddown. For 0 = β0 ≤ β1 ≤ . . . ≤ β` ' n,
12n →
e−β1Ei
Z(β1)→ . . .→ e−β`Ei
Z(β`)
If cooling is too fast, the system can become trapped in a localminimum. Running time depends on the energy landscape.In the worst case, the running time is proportional to the numberof states D = 2n.A quantum simulated annealing algorithm offers a
√· speed-upover all classical annealing processes. (Somma et al.)Used for all sort of (classical) combinatorial optimization problems.These techniques are only suited for classical problems.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 8 / 20
Existing methods for classical ground state
Simulated Annealing
Imitates an initially hot metal (random state) that is slowly cooleddown. For 0 = β0 ≤ β1 ≤ . . . ≤ β` ' n,
12n →
e−β1Ei
Z(β1)→ . . .→ e−β`Ei
Z(β`)
If cooling is too fast, the system can become trapped in a localminimum. Running time depends on the energy landscape.In the worst case, the running time is proportional to the numberof states D = 2n.A quantum simulated annealing algorithm offers a
√· speed-upover all classical annealing processes. (Somma et al.)Used for all sort of (classical) combinatorial optimization problems.These techniques are only suited for classical problems.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 8 / 20
Existing methods for classical ground state
Simulated Annealing
Imitates an initially hot metal (random state) that is slowly cooleddown. For 0 = β0 ≤ β1 ≤ . . . ≤ β` ' n,
12n →
e−β1Ei
Z(β1)→ . . .→ e−β`Ei
Z(β`)
If cooling is too fast, the system can become trapped in a localminimum. Running time depends on the energy landscape.In the worst case, the running time is proportional to the numberof states D = 2n.A quantum simulated annealing algorithm offers a
√· speed-upover all classical annealing processes. (Somma et al.)Used for all sort of (classical) combinatorial optimization problems.These techniques are only suited for classical problems.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 8 / 20
Existing methods for classical ground state
Adiabatic quantum computing
Theorem (Adiabatic evolution)A quantum system will remain in its instantaneous ground state if theHamiltonian that governs its evolution varies slowly enough.
H(t) =tT
Hhard +(T − t)
THeasy
At t = 0, prepare the system in the ground state of Heasy.
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0.
For classical problems, in the worst case the gap is 1/D.(van Dam & Vazirani)No known rigorous lower bound on the gap in the quantum case.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 9 / 20
Existing methods for classical ground state
Adiabatic quantum computing
Theorem (Adiabatic evolution)A quantum system will remain in its instantaneous ground state if theHamiltonian that governs its evolution varies slowly enough.
H(t) =tT
Hhard +(T − t)
THeasy
At t = 0, prepare the system in the ground state of Heasy.
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0.
For classical problems, in the worst case the gap is 1/D.(van Dam & Vazirani)No known rigorous lower bound on the gap in the quantum case.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 9 / 20
Existing methods for classical ground state
Adiabatic quantum computing
Theorem (Adiabatic evolution)A quantum system will remain in its instantaneous ground state if theHamiltonian that governs its evolution varies slowly enough.
H(t) =tT
Hhard +(T − t)
THeasy
At t = 0, prepare the system in the ground state of Heasy.
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0.
For classical problems, in the worst case the gap is 1/D.(van Dam & Vazirani)No known rigorous lower bound on the gap in the quantum case.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 9 / 20
Existing methods for classical ground state
Adiabatic quantum computing
Theorem (Adiabatic evolution)A quantum system will remain in its instantaneous ground state if theHamiltonian that governs its evolution varies slowly enough.
H(t) =tT
Hhard +(T − t)
THeasy
At t = 0, prepare the system in the ground state of Heasy.
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0.
For classical problems, in the worst case the gap is 1/D.(van Dam & Vazirani)No known rigorous lower bound on the gap in the quantum case.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 9 / 20
Existing methods for classical ground state
Adiabatic quantum computing
Theorem (Adiabatic evolution)A quantum system will remain in its instantaneous ground state if theHamiltonian that governs its evolution varies slowly enough.
H(t) =tT
Hhard +(T − t)
THeasy
At t = 0, prepare the system in the ground state of Heasy.
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0.
For classical problems, in the worst case the gap is 1/D.(van Dam & Vazirani)No known rigorous lower bound on the gap in the quantum case.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 9 / 20
Existing methods for classical ground state
Grover’s algorithm
|0! " |1! |0! " |1!!(|0" ! |1"){
H(x) < !|x!
If H(x) ! !
Else
|x!
With |ψ〉 = 1√D
∑i |i〉, we get t ' √D.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 10 / 20
Existing methods for classical ground state
Grover’s algorithm
|0! " |1! |0! " |1!!(|0" ! |1"){
H(x) < !|x!
If H(x) ! !
Else
|x!
|good! = PH<!|!!
|bad! = (1" PH<!)|!!!
With |ψ〉 = 1√D
∑i |i〉, we get t ' √D.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 10 / 20
Existing methods for classical ground state
Grover’s algorithm
|0! " |1! |0! " |1!!(|0" ! |1"){
H(x) < !|x!
If H(x) ! !
Else
|x!
|good! = PH<!|!!
|bad! = (1" PH<!)|!!!!
|!!!
With |ψ〉 = 1√D
∑i |i〉, we get t ' √D.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 10 / 20
Existing methods for classical ground state
Grover’s algorithm
|0! " |1! |0! " |1!!(|0" ! |1"){
H(x) < !|x!
If H(x) ! !
Else
|x!
|good! = PH<!|!!
|bad! = (1" PH<!)|!!!!
2!
With |ψ〉 = 1√D
∑i |i〉, we get t ' √D.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 10 / 20
Existing methods for classical ground state
Grover’s algorithm
|0! " |1! |0! " |1!!(|0" ! |1"){
H(x) < !|x!
If H(x) ! !
Else
|x!
|good! = PH<!|!!
|bad! = (1" PH<!)|!!!!
2!
t ! !
4"! "|good#"!1
With |ψ〉 = 1√D
∑i |i〉, we get t ' √D.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 10 / 20
Existing methods for classical ground state
Grover’s algorithm
|0! " |1! |0! " |1!!(|0" ! |1"){
H(x) < !|x!
If H(x) ! !
Else
|x!
|good! = PH<!|!!
|bad! = (1" PH<!)|!!!!
2!
t ! !
4"! "|good#"!1
With |ψ〉 = 1√D
∑i |i〉, we get t ' √D.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 10 / 20
The obvious method
Outline
1 Motivation
2 Existing methods for classical ground state
3 The obvious method
4 How to fix it...
5 Future work
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 11 / 20
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.Spectral decomposition U|a〉 = ei2πϕa |a〉.
k = polylog(n) “momentum” qubits in the state |0k 〉.
The ’momentum’ ϕa can be measured via Fourier transform.Substituting U = e−iHt with t < ‖H‖−1 provides a method tomeasure the energy.Combining with Grover, we can prepare a state with H < α.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 12 / 20
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.Spectral decomposition U|a〉 = ei2πϕa |a〉.
k = polylog(n) “momentum” qubits in the state |0k 〉.
The ’momentum’ ϕa can be measured via Fourier transform.Substituting U = e−iHt with t < ‖H‖−1 provides a method tomeasure the energy.Combining with Grover, we can prepare a state with H < α.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 12 / 20
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.Spectral decomposition U|a〉 = ei2πϕa |a〉.
k = polylog(n) “momentum” qubits in the state |0k 〉.
|0k!
Ur
H
The ’momentum’ ϕa can be measured via Fourier transform.Substituting U = e−iHt with t < ‖H‖−1 provides a method tomeasure the energy.Combining with Grover, we can prepare a state with H < α.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 12 / 20
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.Spectral decomposition U|a〉 = ei2πϕa |a〉.
k = polylog(n) “momentum” qubits in the state |0k 〉.
|0k!
Ur
H
The ’momentum’ ϕa can be measured via Fourier transform.Substituting U = e−iHt with t < ‖H‖−1 provides a method tomeasure the energy.Combining with Grover, we can prepare a state with H < α.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 12 / 20
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.Spectral decomposition U|a〉 = ei2πϕa |a〉.
k = polylog(n) “momentum” qubits in the state |0k 〉.
|0k!
Ur
H
The ’momentum’ ϕa can be measured via Fourier transform.Substituting U = e−iHt with t < ‖H‖−1 provides a method tomeasure the energy.Combining with Grover, we can prepare a state with H < α.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 12 / 20
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.Spectral decomposition U|a〉 = ei2πϕa |a〉.
k = polylog(n) “momentum” qubits in the state |0k 〉.
|0k!
Ur
H
The ’momentum’ ϕa can be measured via Fourier transform.Substituting U = e−iHt with t < ‖H‖−1 provides a method tomeasure the energy.Combining with Grover, we can prepare a state with H < α.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 12 / 20
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.Spectral decomposition U|a〉 = ei2πϕa |a〉.
k = polylog(n) “momentum” qubits in the state |0k 〉.
|0k!
Ur
H
The ’momentum’ ϕa can be measured via Fourier transform.Substituting U = e−iHt with t < ‖H‖−1 provides a method tomeasure the energy.Combining with Grover, we can prepare a state with H < α.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 12 / 20
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.Spectral decomposition U|a〉 = ei2πϕa |a〉.
k = polylog(n) “momentum” qubits in the state |0k 〉.
|0k!
Ur
H
The ’momentum’ ϕa can be measured via Fourier transform.Substituting U = e−iHt with t < ‖H‖−1 provides a method tomeasure the energy.Combining with Grover, we can prepare a state with H < α.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 12 / 20
The obvious method
Problem: roundoff errors
0 ! 2!
!a = 1.554"
Roundoff errors in the Fourier transform.Grover’s method needs an exact detection of the targeted states:imperfections accumulate.States with energy� α have a small chance of beingmisdiagnosed as having low energy, but there are exponentiallymany of them.For this reason, the success probability of this algorithm isexponentially small.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 13 / 20
The obvious method
Problem: roundoff errors
0 ! 2!
!a = 1.554"
Roundoff errors in the Fourier transform.Grover’s method needs an exact detection of the targeted states:imperfections accumulate.States with energy� α have a small chance of beingmisdiagnosed as having low energy, but there are exponentiallymany of them.For this reason, the success probability of this algorithm isexponentially small.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 13 / 20
The obvious method
Problem: roundoff errors
0 ! 2!
!a = 1.554"
Roundoff errors in the Fourier transform.Grover’s method needs an exact detection of the targeted states:imperfections accumulate.States with energy� α have a small chance of beingmisdiagnosed as having low energy, but there are exponentiallymany of them.For this reason, the success probability of this algorithm isexponentially small.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 13 / 20
The obvious method
Problem: roundoff errors
0 ! 2!
!a = 1.554"
Roundoff errors in the Fourier transform.Grover’s method needs an exact detection of the targeted states:imperfections accumulate.States with energy� α have a small chance of beingmisdiagnosed as having low energy, but there are exponentiallymany of them.For this reason, the success probability of this algorithm isexponentially small.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 13 / 20
The obvious method
Problem: roundoff errors
0 ! 2!
!a = 1.554"
Roundoff errors in the Fourier transform.Grover’s method needs an exact detection of the targeted states:imperfections accumulate.States with energy� α have a small chance of beingmisdiagnosed as having low energy, but there are exponentiallymany of them.For this reason, the success probability of this algorithm isexponentially small.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 13 / 20
How to fix it...
Outline
1 Motivation
2 Existing methods for classical ground state
3 The obvious method
4 How to fix it...
5 Future work
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 14 / 20
How to fix it...
Running the algorithm backward
Failed methodPrepare
∑a µa|a〉 ⊗ |0k 〉.
Use phase estimation to get∑
a µa|a〉 ⊗ |ϕa〉.Amplify the low-momentum states ϕa ≤ α using an imperfectmeasurement.
Better methodPrepare
∑a µa|a〉 ⊗ |ω〉 for some ω ≤ α.
Use inverse phase estimation to get∑
a µa〈ϕa|ω〉|a〉 ⊗ |0k 〉+ . . ..Amplify the state |0k 〉 of the auxiliary qubits using Grover’salgorithm.
This has the effect of ’filtering’ the states by multiplying theiramplitudes by 〈ϕa|ω〉.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 15 / 20
How to fix it...
Running the algorithm backward
Failed methodPrepare
∑a µa|a〉 ⊗ |0k 〉.
Use phase estimation to get∑
a µa|a〉 ⊗ |ϕa〉.Amplify the low-momentum states ϕa ≤ α using an imperfectmeasurement.
Better methodPrepare
∑a µa|a〉 ⊗ |ω〉 for some ω ≤ α.
Use inverse phase estimation to get∑
a µa〈ϕa|ω〉|a〉 ⊗ |0k 〉+ . . ..Amplify the state |0k 〉 of the auxiliary qubits using Grover’salgorithm.
This has the effect of ’filtering’ the states by multiplying theiramplitudes by 〈ϕa|ω〉.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 15 / 20
How to fix it...
Running the algorithm backward
Failed methodPrepare
∑a µa|a〉 ⊗ |0k 〉.
Use phase estimation to get∑
a µa|a〉 ⊗ |ϕa〉.Amplify the low-momentum states ϕa ≤ α using an imperfectmeasurement.
Better methodPrepare
∑a µa|a〉 ⊗ |ω〉 for some ω ≤ α.
Use inverse phase estimation to get∑
a µa〈ϕa|ω〉|a〉 ⊗ |0k 〉+ . . ..Amplify the state |0k 〉 of the auxiliary qubits using Grover’salgorithm.
This has the effect of ’filtering’ the states by multiplying theiramplitudes by 〈ϕa|ω〉.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 15 / 20
How to fix it...
Running the algorithm backward
Failed methodPrepare
∑a µa|a〉 ⊗ |0k 〉.
Use phase estimation to get∑
a µa|a〉 ⊗ |ϕa〉.Amplify the low-momentum states ϕa ≤ α using an imperfectmeasurement.
Better methodPrepare
∑a µa|a〉 ⊗ |ω〉 for some ω ≤ α.
Use inverse phase estimation to get∑
a µa〈ϕa|ω〉|a〉 ⊗ |0k 〉+ . . ..Amplify the state |0k 〉 of the auxiliary qubits using Grover’salgorithm.
This has the effect of ’filtering’ the states by multiplying theiramplitudes by 〈ϕa|ω〉.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 15 / 20
How to fix it...
Running the algorithm backward
Failed methodPrepare
∑a µa|a〉 ⊗ |0k 〉.
Use phase estimation to get∑
a µa|a〉 ⊗ |ϕa〉.Amplify the low-momentum states ϕa ≤ α using an imperfectmeasurement.
Better methodPrepare
∑a µa|a〉 ⊗ |ω〉 for some ω ≤ α.
Use inverse phase estimation to get∑
a µa〈ϕa|ω〉|a〉 ⊗ |0k 〉+ . . ..Amplify the state |0k 〉 of the auxiliary qubits using Grover’salgorithm.
This has the effect of ’filtering’ the states by multiplying theiramplitudes by 〈ϕa|ω〉.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 15 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
How to fix it...
A better filter
Substituted an imperfect measurement (energy measurement ϕa) byan exact measurement (namely of the state |0k 〉).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Summary (Poulin and Wocjan, PRL 2009)Our algorithm prepares a state that lies in the E ± ε spectrum of H.
Running time polylog(D)ε−1√
DR E+εE−ε D(E ′)dE ′
.
By ’scanning’ the values of E and ε, we can prepare the groundstate with little overhead.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 16 / 20
Future work
Outline
1 Motivation
2 Existing methods for classical ground state
3 The obvious method
4 How to fix it...
5 Future work
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 17 / 20
Future work
Efficient algorithms
Non-glassy classical systems often have efficient (heuristic)algorithms.Can we design quantum algorithms that exploit the structure (gap,locality, short-range correlations etc.) of quantum systems toachieve better scalings?
Algorithms to: (Poulin and Wocjan, in prep.)Evaluate partition function Z(β) = Tr(e−βH).
Prepare Gibbs state e−βH
Z(β) .
Prepare quantum Gibbs state∑
a
√e−βEa
Z(β) |a〉 ⊗ |a〉.Running time t = Dα where:
α = h(β)/ log(2) ≤ 1/2h(β) it the Helmholtz free energy density (relative to ground state).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 18 / 20
Future work
Efficient algorithms
Non-glassy classical systems often have efficient (heuristic)algorithms.Can we design quantum algorithms that exploit the structure (gap,locality, short-range correlations etc.) of quantum systems toachieve better scalings?
Algorithms to: (Poulin and Wocjan, in prep.)Evaluate partition function Z(β) = Tr(e−βH).
Prepare Gibbs state e−βH
Z(β) .
Prepare quantum Gibbs state∑
a
√e−βEa
Z(β) |a〉 ⊗ |a〉.Running time t = Dα where:
α = h(β)/ log(2) ≤ 1/2h(β) it the Helmholtz free energy density (relative to ground state).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 18 / 20
Future work
Efficient algorithms
Non-glassy classical systems often have efficient (heuristic)algorithms.Can we design quantum algorithms that exploit the structure (gap,locality, short-range correlations etc.) of quantum systems toachieve better scalings?
Algorithms to: (Poulin and Wocjan, in prep.)Evaluate partition function Z(β) = Tr(e−βH).
Prepare Gibbs state e−βH
Z(β) .
Prepare quantum Gibbs state∑
a
√e−βEa
Z(β) |a〉 ⊗ |a〉.Running time t = Dα where:
α = h(β)/ log(2) ≤ 1/2h(β) it the Helmholtz free energy density (relative to ground state).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 18 / 20
Future work
Efficient algorithms
Non-glassy classical systems often have efficient (heuristic)algorithms.Can we design quantum algorithms that exploit the structure (gap,locality, short-range correlations etc.) of quantum systems toachieve better scalings?
Algorithms to: (Poulin and Wocjan, in prep.)Evaluate partition function Z(β) = Tr(e−βH).
Prepare Gibbs state e−βH
Z(β) .
Prepare quantum Gibbs state∑
a
√e−βEa
Z(β) |a〉 ⊗ |a〉.Running time t = Dα where:
α = h(β)/ log(2) ≤ 1/2h(β) it the Helmholtz free energy density (relative to ground state).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 18 / 20
Future work
Efficient algorithms
Non-glassy classical systems often have efficient (heuristic)algorithms.Can we design quantum algorithms that exploit the structure (gap,locality, short-range correlations etc.) of quantum systems toachieve better scalings?
Algorithms to: (Poulin and Wocjan, in prep.)Evaluate partition function Z(β) = Tr(e−βH).
Prepare Gibbs state e−βH
Z(β) .
Prepare quantum Gibbs state∑
a
√e−βEa
Z(β) |a〉 ⊗ |a〉.Running time t = Dα where:
α = h(β)/ log(2) ≤ 1/2h(β) it the Helmholtz free energy density (relative to ground state).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 18 / 20
Future work
Efficient algorithms
Non-glassy classical systems often have efficient (heuristic)algorithms.Can we design quantum algorithms that exploit the structure (gap,locality, short-range correlations etc.) of quantum systems toachieve better scalings?
Algorithms to: (Poulin and Wocjan, in prep.)Evaluate partition function Z(β) = Tr(e−βH).
Prepare Gibbs state e−βH
Z(β) .
Prepare quantum Gibbs state∑
a
√e−βEa
Z(β) |a〉 ⊗ |a〉.Running time t = Dα where:
α = h(β)/ log(2) ≤ 1/2h(β) it the Helmholtz free energy density (relative to ground state).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 18 / 20
Future work
Efficient algorithms
Non-glassy classical systems often have efficient (heuristic)algorithms.Can we design quantum algorithms that exploit the structure (gap,locality, short-range correlations etc.) of quantum systems toachieve better scalings?
Algorithms to: (Poulin and Wocjan, in prep.)Evaluate partition function Z(β) = Tr(e−βH).
Prepare Gibbs state e−βH
Z(β) .
Prepare quantum Gibbs state∑
a
√e−βEa
Z(β) |a〉 ⊗ |a〉.Running time t = Dα where:
α = h(β)/ log(2) ≤ 1/2h(β) it the Helmholtz free energy density (relative to ground state).
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 18 / 20
Future work
Efficient algorithms: Ising model
H = −∑
i
gσ(i)x + σ
(i)z σ
(i+1)z
!"!!
!""
!"!
"
"#!
"#$
"#%
"#&
"#'
()!*$
()!
()"#"!
+,(+-.
!
!
High T approximation: ! =4"#
t = 2!n
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 19 / 20
Conclusion
There are many connections between quantum information andmany-body physics.Preparing low-energy states is a difficult problem even for aquantum computer.A quantum computer can serve as an efficient universal simulator.We have an algorithm which in the worst case requires
√D steps
(compared to e.g. Lanczos D).Extension of Grover’s algorithm: amplification constrained to asubspace.
We are trying to improve this scaling using special properties ofthe system.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 20 / 20
Conclusion
There are many connections between quantum information andmany-body physics.Preparing low-energy states is a difficult problem even for aquantum computer.A quantum computer can serve as an efficient universal simulator.We have an algorithm which in the worst case requires
√D steps
(compared to e.g. Lanczos D).Extension of Grover’s algorithm: amplification constrained to asubspace.
We are trying to improve this scaling using special properties ofthe system.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 20 / 20
Conclusion
There are many connections between quantum information andmany-body physics.Preparing low-energy states is a difficult problem even for aquantum computer.A quantum computer can serve as an efficient universal simulator.We have an algorithm which in the worst case requires
√D steps
(compared to e.g. Lanczos D).Extension of Grover’s algorithm: amplification constrained to asubspace.
We are trying to improve this scaling using special properties ofthe system.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 20 / 20
Conclusion
There are many connections between quantum information andmany-body physics.Preparing low-energy states is a difficult problem even for aquantum computer.A quantum computer can serve as an efficient universal simulator.We have an algorithm which in the worst case requires
√D steps
(compared to e.g. Lanczos D).Extension of Grover’s algorithm: amplification constrained to asubspace.
We are trying to improve this scaling using special properties ofthe system.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 20 / 20
Conclusion
There are many connections between quantum information andmany-body physics.Preparing low-energy states is a difficult problem even for aquantum computer.A quantum computer can serve as an efficient universal simulator.We have an algorithm which in the worst case requires
√D steps
(compared to e.g. Lanczos D).Extension of Grover’s algorithm: amplification constrained to asubspace.
We are trying to improve this scaling using special properties ofthe system.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 20 / 20
Conclusion
There are many connections between quantum information andmany-body physics.Preparing low-energy states is a difficult problem even for aquantum computer.A quantum computer can serve as an efficient universal simulator.We have an algorithm which in the worst case requires
√D steps
(compared to e.g. Lanczos D).Extension of Grover’s algorithm: amplification constrained to asubspace.
We are trying to improve this scaling using special properties ofthe system.
David Poulin (Sherbrooke) Quantum Thermodynamics APS’09 20 / 20
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