Download Preparing Ground States of Quantum Many

Preparing Ground States of Quantum Many-Body
Systems on a Quantum Computer
David Poulin
Département de Physique
Université de Sherbrooke
Canadian Mathematical Society
Ottawa, December 2008
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
1 / 19
Motivation
Outline
1
Motivation
2
Existing methods
3
The obvious method
4
An algorithm to prepare ground states in time
5
Applications
David Poulin (Sherbrooke)
Preparing Ground States
√
2n
CMS’08
2 / 19
Motivation
Quantum simulation
Many-body problem
Hilbert space: H = (C2 )⊗n .
P
Hamiltonian: H = hi,ji hij where Hermitian hij is the identity
operator except on the ith and jth tensor factor.
Dynamics: ψ ∈ H,
d
dt ψ
= −iHψ. Solution ψ(t) = e−iHt ψ(0).
Ground state: eigenstate associated to lowest eigenvalue of H.
Efficient simulation with a quantum computer.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
3 / 19
Motivation
Quantum simulation
Many-body problem
Hilbert space: H = (C2 )⊗n .
P
Hamiltonian: H = hi,ji hij where Hermitian hij is the identity
operator except on the ith and jth tensor factor.
Dynamics: ψ ∈ H,
d
dt ψ
= −iHψ. Solution ψ(t) = e−iHt ψ(0).
Ground state: eigenstate associated to lowest eigenvalue of H.
Efficient simulation with a quantum computer.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
3 / 19
Motivation
Quantum simulation
Many-body problem
Hilbert space: H = (C2 )⊗n .
P
Hamiltonian: H = hi,ji hij where Hermitian hij is the identity
operator except on the ith and jth tensor factor.
Dynamics: ψ ∈ H,
d
dt ψ
= −iHψ. Solution ψ(t) = e−iHt ψ(0).
Ground state: eigenstate associated to lowest eigenvalue of H.
Efficient simulation with a quantum computer.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
3 / 19
Motivation
Quantum simulation
Many-body problem
Hilbert space: H = (C2 )⊗n .
P
Hamiltonian: H = hi,ji hij where Hermitian hij is the identity
operator except on the ith and jth tensor factor.
Dynamics: ψ ∈ H,
d
dt ψ
= −iHψ. Solution ψ(t) = e−iHt ψ(0).
Ground state: eigenstate associated to lowest eigenvalue of H.
Efficient simulation with a quantum computer.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
3 / 19
Motivation
Quantum simulation
Many-body problem
Hilbert space: H = (C2 )⊗n .
P
Hamiltonian: H = hi,ji hij where Hermitian hij is the identity
operator except on the ith and jth tensor factor.
Dynamics: ψ ∈ H,
d
dt ψ
= −iHψ. Solution ψ(t) = e−iHt ψ(0).
Ground state: eigenstate associated to lowest eigenvalue of H.
Efficient simulation with a quantum computer.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
3 / 19
Motivation
Quantum simulation
Many-body problem
Hilbert space: H = (C2 )⊗n .
P
Hamiltonian: H = hi,ji hij where Hermitian hij is the identity
operator except on the ith and jth tensor factor.
Dynamics: ψ ∈ H,
d
dt ψ
= −iHψ. Solution ψ(t) = e−iHt ψ(0).
Ground state: eigenstate associated to lowest eigenvalue of H.
Efficient simulation with a quantum computer.
A
e-iHt =
U
# of gates = poly(n,t)
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
3 / 19
Motivation
A notoriously hard problem
This ’integration tool’ is not of much use if we cannot specify the
initial conditions; what is ψ(0)?
Physically relevant are the low energy states of H.
Preparing them is a difficult problem:
Question
Is the lowest eigenvalue of H less than α or greater than β where
β − α > 1/poly (n).
Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
4 / 19
Motivation
A notoriously hard problem
This ’integration tool’ is not of much use if we cannot specify the
initial conditions; what is ψ(0)?
Physically relevant are the low energy states of H.
Preparing them is a difficult problem:
Question
Is the lowest eigenvalue of H less than α or greater than β where
β − α > 1/poly (n).
Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
4 / 19
Motivation
A notoriously hard problem
This ’integration tool’ is not of much use if we cannot specify the
initial conditions; what is ψ(0)?
Physically relevant are the low energy states of H.
Preparing them is a difficult problem:
Question
Is the lowest eigenvalue of H less than α or greater than β where
β − α > 1/poly (n).
Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
4 / 19
Motivation
A notoriously hard problem
This ’integration tool’ is not of much use if we cannot specify the
initial conditions; what is ψ(0)?
Physically relevant are the low energy states of H.
Preparing them is a difficult problem:
Question
Is the lowest eigenvalue of H less than α or greater than β where
β − α > 1/poly (n).
Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
4 / 19
Motivation
A notoriously hard problem
This ’integration tool’ is not of much use if we cannot specify the
initial conditions; what is ψ(0)?
Physically relevant are the low energy states of H.
Preparing them is a difficult problem:
Question
Is the lowest eigenvalue of H less than α or greater than β where
β − α > 1/poly (n).
Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona).
Quantum case is QMA-complete (Kitaev).
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
4 / 19
Existing methods
Outline
1
Motivation
2
Existing methods
3
The obvious method
4
An algorithm to prepare ground states in time
5
Applications
David Poulin (Sherbrooke)
Preparing Ground States
√
2n
CMS’08
5 / 19
Existing methods
Simulated Annealing
Used for all sort of (classical) combinatorial optimization problems.
Imitates an initially hot metal (random state) that is slowly cooled
down.
If cooling is too fast, the system can become trapped in a local
minimum. Running time depends on the energy landscape.
In the worst case, the running time is proportional to the number
of states N = 2n .
√
A quantum simulated annealing algorithm offers a · speed-up
over all classical annealing processes. (Somma et al.)
These techniques are only suited for classical problems.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
6 / 19
Existing methods
Simulated Annealing
Used for all sort of (classical) combinatorial optimization problems.
Imitates an initially hot metal (random state) that is slowly cooled
down.
If cooling is too fast, the system can become trapped in a local
minimum. Running time depends on the energy landscape.
In the worst case, the running time is proportional to the number
of states N = 2n .
√
A quantum simulated annealing algorithm offers a · speed-up
over all classical annealing processes. (Somma et al.)
These techniques are only suited for classical problems.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
6 / 19
Existing methods
Simulated Annealing
Used for all sort of (classical) combinatorial optimization problems.
Imitates an initially hot metal (random state) that is slowly cooled
down.
If cooling is too fast, the system can become trapped in a local
minimum. Running time depends on the energy landscape.
In the worst case, the running time is proportional to the number
of states N = 2n .
√
A quantum simulated annealing algorithm offers a · speed-up
over all classical annealing processes. (Somma et al.)
These techniques are only suited for classical problems.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
6 / 19
Existing methods
Simulated Annealing
Used for all sort of (classical) combinatorial optimization problems.
Imitates an initially hot metal (random state) that is slowly cooled
down.
If cooling is too fast, the system can become trapped in a local
minimum. Running time depends on the energy landscape.
In the worst case, the running time is proportional to the number
of states N = 2n .
√
A quantum simulated annealing algorithm offers a · speed-up
over all classical annealing processes. (Somma et al.)
These techniques are only suited for classical problems.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
6 / 19
Existing methods
Simulated Annealing
Used for all sort of (classical) combinatorial optimization problems.
Imitates an initially hot metal (random state) that is slowly cooled
down.
If cooling is too fast, the system can become trapped in a local
minimum. Running time depends on the energy landscape.
In the worst case, the running time is proportional to the number
of states N = 2n .
√
A quantum simulated annealing algorithm offers a · speed-up
over all classical annealing processes. (Somma et al.)
These techniques are only suited for classical problems.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
6 / 19
Existing methods
Simulated Annealing
Used for all sort of (classical) combinatorial optimization problems.
Imitates an initially hot metal (random state) that is slowly cooled
down.
If cooling is too fast, the system can become trapped in a local
minimum. Running time depends on the energy landscape.
In the worst case, the running time is proportional to the number
of states N = 2n .
√
A quantum simulated annealing algorithm offers a · speed-up
over all classical annealing processes. (Somma et al.)
These techniques are only suited for classical problems.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
6 / 19
Existing methods
Adiabatic quantum computing
Theorem (Adiabatic evolution)
A quantum system will remain in its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough.
(T − t)
t
Hhard +
Heasy
T
T
At t = 0, prepare the system in the ground state of Heasy .
H(t) =
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 .
For classical problems, in the worst case the gap is 1/2n .
(van Dam & Vazirani)
Not studied for the quantum case.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
7 / 19
Existing methods
Adiabatic quantum computing
Theorem (Adiabatic evolution)
A quantum system will remain in its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough.
(T − t)
t
Hhard +
Heasy
T
T
At t = 0, prepare the system in the ground state of Heasy .
H(t) =
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 .
For classical problems, in the worst case the gap is 1/2n .
(van Dam & Vazirani)
Not studied for the quantum case.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
7 / 19
Existing methods
Adiabatic quantum computing
Theorem (Adiabatic evolution)
A quantum system will remain in its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough.
(T − t)
t
Hhard +
Heasy
T
T
At t = 0, prepare the system in the ground state of Heasy .
H(t) =
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 .
For classical problems, in the worst case the gap is 1/2n .
(van Dam & Vazirani)
Not studied for the quantum case.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
7 / 19
Existing methods
Adiabatic quantum computing
Theorem (Adiabatic evolution)
A quantum system will remain in its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough.
(T − t)
t
Hhard +
Heasy
T
T
At t = 0, prepare the system in the ground state of Heasy .
H(t) =
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 .
For classical problems, in the worst case the gap is 1/2n .
(van Dam & Vazirani)
Not studied for the quantum case.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
7 / 19
Existing methods
Adiabatic quantum computing
Theorem (Adiabatic evolution)
A quantum system will remain in its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough.
(T − t)
t
Hhard +
Heasy
T
T
At t = 0, prepare the system in the ground state of Heasy .
H(t) =
Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 .
For classical problems, in the worst case the gap is 1/2n .
(van Dam & Vazirani)
Not studied for the quantum case.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
7 / 19
Existing methods
Grover’s algorithm
Given a projector P onto a subspace of H and a state ψ, construct the
two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|.
Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1
2 e.
Black box interpretation
Maps state |ψi to
1
kP|ψik P|ψi
in time
1
kP|ψik .
Choose P to be the projection on the ’solution’, i.e. onto H < α.
q
−1
Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) .
√
Then the solution is found in t ≤ 2n iterations.
The projector onto H < α is easy to construct for classical H.
How to construct it for quantum systems?
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
8 / 19
Existing methods
Grover’s algorithm
Given a projector P onto a subspace of H and a state ψ, construct the
two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|.
Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1
2 e.
Black box interpretation
Maps state |ψi to
1
kP|ψik P|ψi
in time
1
kP|ψik .
Choose P to be the projection on the ’solution’, i.e. onto H < α.
q
−1
Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) .
√
Then the solution is found in t ≤ 2n iterations.
The projector onto H < α is easy to construct for classical H.
How to construct it for quantum systems?
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
8 / 19
Existing methods
Grover’s algorithm
Given a projector P onto a subspace of H and a state ψ, construct the
two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|.
Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1
2 e.
Black box interpretation
Maps state |ψi to
1
kP|ψik P|ψi
in time
1
kP|ψik .
Choose P to be the projection on the ’solution’, i.e. onto H < α.
q
−1
Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) .
√
Then the solution is found in t ≤ 2n iterations.
The projector onto H < α is easy to construct for classical H.
How to construct it for quantum systems?
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
8 / 19
Existing methods
Grover’s algorithm
Given a projector P onto a subspace of H and a state ψ, construct the
two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|.
Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1
2 e.
Black box interpretation
Maps state |ψi to
1
kP|ψik P|ψi
in time
1
kP|ψik .
Choose P to be the projection on the ’solution’, i.e. onto H < α.
q
−1
Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) .
√
Then the solution is found in t ≤ 2n iterations.
The projector onto H < α is easy to construct for classical H.
How to construct it for quantum systems?
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
8 / 19
Existing methods
Grover’s algorithm
Given a projector P onto a subspace of H and a state ψ, construct the
two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|.
Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1
2 e.
Black box interpretation
Maps state |ψi to
1
kP|ψik P|ψi
in time
1
kP|ψik .
Choose P to be the projection on the ’solution’, i.e. onto H < α.
q
−1
Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) .
√
Then the solution is found in t ≤ 2n iterations.
The projector onto H < α is easy to construct for classical H.
How to construct it for quantum systems?
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
8 / 19
Existing methods
Grover’s algorithm
Given a projector P onto a subspace of H and a state ψ, construct the
two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|.
Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1
2 e.
Black box interpretation
Maps state |ψi to
1
kP|ψik P|ψi
in time
1
kP|ψik .
Choose P to be the projection on the ’solution’, i.e. onto H < α.
q
−1
Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) .
√
Then the solution is found in t ≤ 2n iterations.
The projector onto H < α is easy to construct for classical H.
How to construct it for quantum systems?
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
8 / 19
Existing methods
Grover’s algorithm
Given a projector P onto a subspace of H and a state ψ, construct the
two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|.
Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1
2 e.
Black box interpretation
Maps state |ψi to
1
kP|ψik P|ψi
in time
1
kP|ψik .
Choose P to be the projection on the ’solution’, i.e. onto H < α.
q
−1
Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) .
√
Then the solution is found in t ≤ 2n iterations.
The projector onto H < α is easy to construct for classical H.
How to construct it for quantum systems?
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
8 / 19
Existing methods
Grover’s algorithm
Given a projector P onto a subspace of H and a state ψ, construct the
two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|.
Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1
2 e.
Black box interpretation
Maps state |ψi to
1
kP|ψik P|ψi
in time
1
kP|ψik .
Choose P to be the projection on the ’solution’, i.e. onto H < α.
q
−1
Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) .
√
Then the solution is found in t ≤ 2n iterations.
The projector onto H < α is easy to construct for classical H.
How to construct it for quantum systems?
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
8 / 19
The obvious method
Outline
1
Motivation
2
Existing methods
3
The obvious method
4
An algorithm to prepare ground states in time
5
Applications
David Poulin (Sherbrooke)
Preparing Ground States
√
2n
CMS’08
9 / 19
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.
Spectral decomposition U|ai = ei2πϕa |ai.
k = polylog(n) auxiliary qubits in the state |0k i.
Phase estimation maps |ai ⊗ |0k i → |ai ⊗ |ϕa i in poly time where
X
|ϕa i =
ei2πϕa j |ji.
j
These are ’momentum’ states so the value of ϕa can be measured
via Fourier transform.
Substituting U = e−iHt with t < kHk−1 provides a method to
measure the energy and hence to construct a projector onto
H < α for use in Grover’s algorithm.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
10 / 19
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.
Spectral decomposition U|ai = ei2πϕa |ai.
k = polylog(n) auxiliary qubits in the state |0k i.
Phase estimation maps |ai ⊗ |0k i → |ai ⊗ |ϕa i in poly time where
X
|ϕa i =
ei2πϕa j |ji.
j
These are ’momentum’ states so the value of ϕa can be measured
via Fourier transform.
Substituting U = e−iHt with t < kHk−1 provides a method to
measure the energy and hence to construct a projector onto
H < α for use in Grover’s algorithm.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
10 / 19
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.
Spectral decomposition U|ai = ei2πϕa |ai.
k = polylog(n) auxiliary qubits in the state |0k i.
Phase estimation maps |ai ⊗ |0k i → |ai ⊗ |ϕa i in poly time where
X
|ϕa i =
ei2πϕa j |ji.
j
These are ’momentum’ states so the value of ϕa can be measured
via Fourier transform.
Substituting U = e−iHt with t < kHk−1 provides a method to
measure the energy and hence to construct a projector onto
H < α for use in Grover’s algorithm.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
10 / 19
The obvious method
Phase estimation
Ingredients
An efficiently implementable unitary matrix U on n qubits.
Spectral decomposition U|ai = ei2πϕa |ai.
k = polylog(n) auxiliary qubits in the state |0k i.
Phase estimation maps |ai ⊗ |0k i → |ai ⊗ |ϕa i in poly time where
X
|ϕa i =
ei2πϕa j |ji.
j
These are ’momentum’ states so the value of ϕa can be measured
via Fourier transform.
Substituting U = e−iHt with t < kHk−1 provides a method to
measure the energy and hence to construct a projector onto
H < α for use in Grover’s algorithm.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
10 / 19
The obvious method
Problem
This is an approximate method to measure energy and errors build up
during Grover’s algorithm.
The unitary U = e−itH can only be approximated by the QC.
Round-off errors in the Fourier transform. There is a small
probability that an eigenstate |ai with phase ϕa << α be
diagnoses as having a phase >> α.
Let P be the projector onto the accepting subspace of the phase
estimation algorithm.
p
√
|ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i
Using Jordan’s block decomposition, we find that
p
√
|gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥
ki
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
11 / 19
The obvious method
Problem
This is an approximate method to measure energy and errors build up
during Grover’s algorithm.
The unitary U = e−itH can only be approximated by the QC.
Round-off errors in the Fourier transform. There is a small
probability that an eigenstate |ai with phase ϕa << α be
diagnoses as having a phase >> α.
Let P be the projector onto the accepting subspace of the phase
estimation algorithm.
p
√
|ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i
Using Jordan’s block decomposition, we find that
p
√
|gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥
ki
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
11 / 19
The obvious method
Problem
This is an approximate method to measure energy and errors build up
during Grover’s algorithm.
The unitary U = e−itH can only be approximated by the QC.
Round-off errors in the Fourier transform. There is a small
probability that an eigenstate |ai with phase ϕa << α be
diagnoses as having a phase >> α.
Let P be the projector onto the accepting subspace of the phase
estimation algorithm.
p
√
|ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i
Using Jordan’s block decomposition, we find that
p
√
|gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥
ki
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
11 / 19
The obvious method
Problem
This is an approximate method to measure energy and errors build up
during Grover’s algorithm.
The unitary U = e−itH can only be approximated by the QC.
Round-off errors in the Fourier transform. There is a small
probability that an eigenstate |ai with phase ϕa << α be
diagnoses as having a phase >> α.
Let P be the projector onto the accepting subspace of the phase
estimation algorithm.
p
√
|ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i
Using Jordan’s block decomposition, we find that
p
√
|gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥
ki
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
11 / 19
The obvious method
Problem
This is an approximate method to measure energy and errors build up
during Grover’s algorithm.
The unitary U = e−itH can only be approximated by the QC.
Round-off errors in the Fourier transform. There is a small
probability that an eigenstate |ai with phase ϕa << α be
diagnoses as having a phase >> α.
Let P be the projector onto the accepting subspace of the phase
estimation algorithm.
p
√
|ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i
Using Jordan’s block decomposition, we find that
p
√
|gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥
ki
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
11 / 19
The obvious method
Problem
So let’s see what happens when we apply Grover’s algorithm to a
random state...
X
|ψi =
µa |ai ⊗ |0k i
X √
p
=
µa pa |gooda i + 1 − pa |bada i
1 X √
Grover
−−−−→
µa pa |gooda i
kPψk
p
1 X
=
µa (pa |ai ⊗ |0k i − pa (1 − pa )|ai ⊗ |0⊥
k i)
kPψk
We have entangled the system with the auxiliary qubits and the vast
majority of the amplitude is on state with incorrect auxiliary qubits
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
12 / 19
The obvious method
Problem
So let’s see what happens when we apply Grover’s algorithm to a
random state...
X
|ψi =
µa |ai ⊗ |0k i
X √
p
=
µa pa |gooda i + 1 − pa |bada i
1 X √
Grover
−−−−→
µa pa |gooda i
kPψk
p
1 X
=
µa (pa |ai ⊗ |0k i − pa (1 − pa )|ai ⊗ |0⊥
k i)
kPψk
We have entangled the system with the auxiliary qubits and the vast
majority of the amplitude is on state with incorrect auxiliary qubits
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
12 / 19
The obvious method
Problem
So let’s see what happens when we apply Grover’s algorithm to a
random state...
X
|ψi =
µa |ai ⊗ |0k i
X √
p
=
µa pa |gooda i + 1 − pa |bada i
1 X √
Grover
−−−−→
µa pa |gooda i
kPψk
p
1 X
=
µa (pa |ai ⊗ |0k i − pa (1 − pa )|ai ⊗ |0⊥
k i)
kPψk
We have entangled the system with the auxiliary qubits and the vast
majority of the amplitude is on state with incorrect auxiliary qubits
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
12 / 19
An algorithm to prepare ground states in time
√
2n
Outline
1
Motivation
2
Existing methods
3
The obvious method
4
An algorithm to prepare ground states in time
5
Applications
David Poulin (Sherbrooke)
Preparing Ground States
√
2n
CMS’08
13 / 19
An algorithm to prepare ground states in time
√
2n
Running the algorithm backward
Failed method
P
Prepare a µa |ai ⊗ |0k i.
Use phase estimation to get
P
a µa |ai
⊗ |ϕa i.
Amplify the low-momentum states ϕa ≤ α using an imperfect
measurement.
Better method
P
Prepare a µa |ai ⊗ |ωi for some ω ≤ α.
P
Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . ..
Amplifie the state |0k i of the auxiliary qubits.
hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) .
This procedure thus filters out states whose energy is not near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
14 / 19
An algorithm to prepare ground states in time
√
2n
Running the algorithm backward
Failed method
P
Prepare a µa |ai ⊗ |0k i.
Use phase estimation to get
P
a µa |ai
⊗ |ϕa i.
Amplify the low-momentum states ϕa ≤ α using an imperfect
measurement.
Better method
P
Prepare a µa |ai ⊗ |ωi for some ω ≤ α.
P
Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . ..
Amplifie the state |0k i of the auxiliary qubits.
hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) .
This procedure thus filters out states whose energy is not near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
14 / 19
An algorithm to prepare ground states in time
√
2n
Running the algorithm backward
Failed method
P
Prepare a µa |ai ⊗ |0k i.
Use phase estimation to get
P
a µa |ai
⊗ |ϕa i.
Amplify the low-momentum states ϕa ≤ α using an imperfect
measurement.
Better method
P
Prepare a µa |ai ⊗ |ωi for some ω ≤ α.
P
Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . ..
Amplifie the state |0k i of the auxiliary qubits.
hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) .
This procedure thus filters out states whose energy is not near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
14 / 19
An algorithm to prepare ground states in time
√
2n
Running the algorithm backward
Failed method
P
Prepare a µa |ai ⊗ |0k i.
Use phase estimation to get
P
a µa |ai
⊗ |ϕa i.
Amplify the low-momentum states ϕa ≤ α using an imperfect
measurement.
Better method
P
Prepare a µa |ai ⊗ |ωi for some ω ≤ α.
P
Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . ..
Amplifie the state |0k i of the auxiliary qubits.
hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) .
This procedure thus filters out states whose energy is not near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
14 / 19
An algorithm to prepare ground states in time
√
2n
Running the algorithm backward
Failed method
P
Prepare a µa |ai ⊗ |0k i.
Use phase estimation to get
P
a µa |ai
⊗ |ϕa i.
Amplify the low-momentum states ϕa ≤ α using an imperfect
measurement.
Better method
P
Prepare a µa |ai ⊗ |ωi for some ω ≤ α.
P
Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . ..
Amplifie the state |0k i of the auxiliary qubits.
hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) .
This procedure thus filters out states whose energy is not near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
14 / 19
An algorithm to prepare ground states in time
√
2n
Running the algorithm backward
Failed method
P
Prepare a µa |ai ⊗ |0k i.
Use phase estimation to get
P
a µa |ai
⊗ |ϕa i.
Amplify the low-momentum states ϕa ≤ α using an imperfect
measurement.
Better method
P
Prepare a µa |ai ⊗ |ωi for some ω ≤ α.
P
Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . ..
Amplifie the state |0k i of the auxiliary qubits.
hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) .
This procedure thus filters out states whose energy is not near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
14 / 19
An algorithm to prepare ground states in time
√
2n
A better filter
Our filter has a heavy tail:
|hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|).
Since there are an exponential number of states that can
contribute to the energy, we need an exponential filter.
Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η
With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show:
When there is no eigenvalue of H in the interval ω ± /2, Grover’s
algorithm succeeds with probability ≤ 2−n .
√
If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm
has a constant probability of succeeding.
When Grover’s algorithm succeeds, its output has energy within
ω ± /2 with probability ≥ 1 − 2−n .
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
15 / 19
An algorithm to prepare ground states in time
√
2n
A better filter
Our filter has a heavy tail:
|hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|).
Since there are an exponential number of states that can
contribute to the energy, we need an exponential filter.
Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η
With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show:
When there is no eigenvalue of H in the interval ω ± /2, Grover’s
algorithm succeeds with probability ≤ 2−n .
√
If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm
has a constant probability of succeeding.
When Grover’s algorithm succeeds, its output has energy within
ω ± /2 with probability ≥ 1 − 2−n .
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
15 / 19
An algorithm to prepare ground states in time
√
2n
A better filter
Our filter has a heavy tail:
|hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|).
Since there are an exponential number of states that can
contribute to the energy, we need an exponential filter.
Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η
With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show:
When there is no eigenvalue of H in the interval ω ± /2, Grover’s
algorithm succeeds with probability ≤ 2−n .
√
If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm
has a constant probability of succeeding.
When Grover’s algorithm succeeds, its output has energy within
ω ± /2 with probability ≥ 1 − 2−n .
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
15 / 19
An algorithm to prepare ground states in time
√
2n
A better filter
Our filter has a heavy tail:
|hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|).
Since there are an exponential number of states that can
contribute to the energy, we need an exponential filter.
Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η
With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show:
When there is no eigenvalue of H in the interval ω ± /2, Grover’s
algorithm succeeds with probability ≤ 2−n .
√
If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm
has a constant probability of succeeding.
When Grover’s algorithm succeeds, its output has energy within
ω ± /2 with probability ≥ 1 − 2−n .
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
15 / 19
An algorithm to prepare ground states in time
√
2n
A better filter
Our filter has a heavy tail:
|hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|).
Since there are an exponential number of states that can
contribute to the energy, we need an exponential filter.
Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η
With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show:
When there is no eigenvalue of H in the interval ω ± /2, Grover’s
algorithm succeeds with probability ≤ 2−n .
√
If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm
has a constant probability of succeeding.
When Grover’s algorithm succeeds, its output has energy within
ω ± /2 with probability ≥ 1 − 2−n .
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
15 / 19
An algorithm to prepare ground states in time
√
2n
A better filter
Our filter has a heavy tail:
|hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|).
Since there are an exponential number of states that can
contribute to the energy, we need an exponential filter.
Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η
With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show:
When there is no eigenvalue of H in the interval ω ± /2, Grover’s
algorithm succeeds with probability ≤ 2−n .
√
If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm
has a constant probability of succeeding.
When Grover’s algorithm succeeds, its output has energy within
ω ± /2 with probability ≥ 1 − 2−n .
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
15 / 19
Applications
Outline
1
Motivation
2
Existing methods
3
The obvious method
4
An algorithm to prepare ground states in time
5
Applications
David Poulin (Sherbrooke)
Preparing Ground States
√
2n
CMS’08
16 / 19
Applications
Preparing thermal states
Thermal state at temperature T is
1 −H/T
Ze
where Z = Tr {e−H/T }.
Quantum counting combined
P with previous method to estimate
density of state D(ω) = a δ(ω − ϕa ).
Choose a random ω following the distribution e−ω/T D(ω).
Use the previous algorithm to create a state of energy near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
17 / 19
Applications
Preparing thermal states
Thermal state at temperature T is
1 −H/T
Ze
where Z = Tr {e−H/T }.
Quantum counting combined
P with previous method to estimate
density of state D(ω) = a δ(ω − ϕa ).
Choose a random ω following the distribution e−ω/T D(ω).
Use the previous algorithm to create a state of energy near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
17 / 19
Applications
Preparing thermal states
Thermal state at temperature T is
1 −H/T
Ze
where Z = Tr {e−H/T }.
Quantum counting combined
P with previous method to estimate
density of state D(ω) = a δ(ω − ϕa ).
Choose a random ω following the distribution e−ω/T D(ω).
Use the previous algorithm to create a state of energy near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
17 / 19
Applications
Preparing thermal states
Thermal state at temperature T is
1 −H/T
Ze
where Z = Tr {e−H/T }.
Quantum counting combined
P with previous method to estimate
density of state D(ω) = a δ(ω − ϕa ).
Choose a random ω following the distribution e−ω/T D(ω).
Use the previous algorithm to create a state of energy near ω.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
17 / 19
Applications
Solving QMA
Given an efficient quantum circuit V , maximize the probability of
observing its first output qubit in the state 1 over all input states if
the form |ψn i ⊗ |0h i.
Reduces to local Hamiltonian (Kiatev) but this does not preserve
witness size.
Acceptance probability of V ⇔ Energy of H.
Marriott and Watrous have designed a circuit to estimate the
acceptance probability of V :
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
18 / 19
Applications
Solving QMA
Given an efficient quantum circuit V , maximize the probability of
observing its first output qubit in the state 1 over all input states if
the form |ψn i ⊗ |0h i.
Reduces to local Hamiltonian (Kiatev) but this does not preserve
witness size.
Acceptance probability of V ⇔ Energy of H.
Marriott and Watrous have designed a circuit to estimate the
acceptance probability of V :
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
18 / 19
Applications
Solving QMA
Given an efficient quantum circuit V , maximize the probability of
observing its first output qubit in the state 1 over all input states if
the form |ψn i ⊗ |0h i.
Reduces to local Hamiltonian (Kiatev) but this does not preserve
witness size.
Acceptance probability of V ⇔ Energy of H.
Marriott and Watrous have designed a circuit to estimate the
acceptance probability of V :
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
18 / 19
Applications
Solving QMA
Given an efficient quantum circuit V , maximize the probability of
observing its first output qubit in the state 1 over all input states if
the form |ψn i ⊗ |0h i.
Reduces to local Hamiltonian (Kiatev) but this does not preserve
witness size.
Acceptance probability of V ⇔ Energy of H.
Marriott and Watrous have designed a circuit to estimate the
acceptance probability of V :
V
V
-1
V
-1
V
V
-1
V
|ωi =
David Poulin (Sherbrooke)
P √
j
ω
Preparing Ground States
k −s(w) √
1−ω
s(j)
(−1)`(j) |ji
CMS’08
18 / 19
Summary
Conclusion
For a quantum computer...
Preparing the ground state of a quantum system is no more
difficult than preparing the ground state of a classical system.
QMA is no harder than NP† .
†
Assuming that all problems in QMA require only a log-size scratch pad.
David Poulin (Sherbrooke)
Preparing Ground States
CMS’08
19 / 19