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The Bose-Hubbard model is QMA-complete. Andrew M. Childs David Gosset Zak Webb arXiv : 1311.3297 Institute for Q uantum Computing University of Waterloo. Solving for the ground energy of a quantum system can be viewed as a quantum constraint satisfaction problem. How difficult is it?. - PowerPoint PPT Presentation
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The Bose-Hubbard model is QMA-complete
Andrew M. ChildsDavid Gosset
Zak Webb
arXiv: 1311.3297
Institute for Quantum Computing University of Waterloo
Solving for the ground energy of a quantum system can be viewed as a quantum constraint satisfaction problem. How difficult is it?
Image source: http://www.condmat.physics.manchester.ac.uk/imagelibrary/
QMA1-complete for
Local Ground energy problem
Frustration-free
Stoquastic(no “sign problem”)
Quantum k-SAT (testing frustration-freeness)
k-local Hamiltonian problem
Stoquastic k-local Hamiltonian problem
Fermions or Bosons
Class of Hamiltonians ComplexityQMA-complete for
Contained in AMMA-hard
QMA-complete
[Kempe, Kitaev, Regev 2006]
[Bravyi et. al. 2006]
[Liu, Christandl, Verstraete 2007] [Wei, Mosca, Nayak 2010]
Contained in P for
[Bravyi 2006] [G. , Nagaj 2013 ]
The computational difficulty of computing the ground energy has been studied for many broad classes of Hamiltonians
[Kitaev 1999] [Kempe, Regev 2003]
h 𝑖𝑗
𝐻 𝑖 , 𝑖+1
�⃗�𝑖
2-local Hamiltonian on a 2D grid [Oliveira Terhal 2008]
2-local Hamiltonian on a line with qudits[Aharonov et. al 2009] [Gottesman Irani 2009]
Hubbard model on a 2D grid with site-dependent magnetic field [Schuch Verstraete 2009].
Versions of the XY, Heisenberg, and other models with adjustable coefficients[Cubitt Montanaro 2013] )
E.g.,
Systems with QMA-complete ground energy problems can be surprisingly simple
…However, the complexity of many natural models from condensed matter physics is still not understood, e.g., the XY model on a graph
)
…However, the complexity of many natural models from condensed matter physics is still not understood, e.g., the XY model on a graph
Unfortunately, proof techniques using perturbation theory * require coefficients which grow with system size, e.g., [Cubitt Montanaro 2013]
*[Kempe Kitaev Regev 2006], [Oliveira Terhal 2008]
)
Allowed to scale polynomially with n
)
…However, the complexity of many natural models from condensed matter physics is still not understood, e.g., the XY model on a graph
Unfortunately, proof techniques using perturbation theory * require coefficients which grow with system size, e.g., [Cubitt Montanaro 2013]
In this work we consider a model of interacting Bosons on a graph with no adjustable coefficients…
*[Kempe Kitaev Regev 2006], [Oliveira Terhal 2008]
)
Allowed to scale polynomially with n
)
Bosons move between adjacent vertices and experience an energy penalty if two or more particles occupy the same site.
Adjacency matrix (a symmetric 0-1 matrix)
The Bose-Hubbard model on a graph
Bosons move between adjacent vertices and experience an energy penalty if two or more particles occupy the same site.
𝐻𝐺(𝑡 ,𝑈 )=𝑡 ∑𝑖 , 𝑗 ∈𝑉
𝐴(𝐺)𝑖𝑗 𝑎𝑖† 𝑎 𝑗+𝑈 ∑𝑘∈𝑉
𝑛𝑘 (𝑛𝑘−1 )
Movement Repulsive on-site interaction
Adjacency matrix (a symmetric 0-1 matrix)
Conserves totalnumber of particles
The Bose-Hubbard model on a graph
annihilates a particle at site i counts the number of particles at site i
Bosons move between adjacent vertices and experience an energy penalty if two or more particles occupy the same site.
The Bose-Hubbard model on a graph
𝐻𝐺(𝑡 ,𝑈 )=𝑡 ∑𝑖 , 𝑗 ∈𝑉
𝐴(𝐺)𝑖𝑗 𝑎𝑖† 𝑎 𝑗+𝑈 ∑𝑘∈𝑉
𝑛𝑘 (𝑛𝑘−1 )
Movement Repulsive on-site interaction
Adjacency matrix (a symmetric 0-1 matrix)
(t,U)-Bose-Hubbard Hamiltonian problemInput: A graph number of particles , energy threshold , and precision parameter Problem: Is the ground energy of in the -particle sector at most or at least (promised that one of these conditions holds)
Conserves totalnumber of particles
𝑈
QMA-complete[our results]
𝑡
Theorem: (t,U)-Bose Hubbard Hamiltonian is QMA-complete for all .
Complexity of (t,U)-Bose Hubbard Hamiltonian
𝑈“Stoquastic”
QMA-complete[our results]
Contained in AMQMA[Bravyi et. al 2006]
𝑡
Theorem: (t,U)-Bose Hubbard Hamiltonian is QMA-complete for all .
Complexity of (t,U)-Bose Hubbard Hamiltonian
𝑈“Stoquastic”
Contained in QMA QMA-complete
[our results]
Contained in AMQMA[Bravyi et. al 2006]
𝑡
Theorem: (t,U)-Bose Hubbard Hamiltonian is QMA-complete for all .
Complexity of (t,U)-Bose Hubbard Hamiltonian
Complexity of (t,U)-Bose Hubbard Hamiltonian
In the limit of infinite repulsion (i.e., hard core bosons) there is only ever 0 or 1 particle at each vertex, and the Hamiltonian is equivalent to a spin model…
𝑈“Stoquastic”
Contained in QMA QMA-complete
[our results]
Contained in AMQMA[Bravyi et. al 2006]
𝑡
Theorem: (t,U)-Bose Hubbard Hamiltonian is QMA-complete for all .
A related class of 2-local Hamiltonians
Conserves total magnetization (Hamming weight)
XY Hamiltonian problemInput: A graph , magnetization , energy threshold , precision parameter Problem: Is the ground energy of in the sector with magnetization at most or at least (promised one of these conditions holds)
Theorem: XY Hamiltonian is QMA-complete.
Quantum Merlin Arthur
If is a yes instance there exists (a witness) which is accepted with high probability.If is a no instance every state has low acceptance probability.
Wm-1Wm-2…W0
¿𝜓 ⟩¿0 ⟩⊗𝑛𝑎
Every instance of a problem in QMA has a verification circuit
QMA: class of decision problems where yes instances can be efficiently verified on a quantum computer.
Ground energy problems are usually contained in QMA (witness ground state ; verification circuit energy measurement)
Proving QMA-hardness is more involved…
Proving QMA-hardness for ground energy problems
Wm-1Wm-2…W0¿𝜓 ⟩
¿0 ⟩⊗𝑛𝑎 𝐻 𝑥
Desired properties:Ground energy of is small Verification circuit accepts a state with high probability is a yes instance
QMA Verification circuit for Hamiltonian
Proving QMA-hardness for ground energy problems
Wm-1Wm-2…W0¿𝜓 ⟩
¿0 ⟩⊗𝑛𝑎 𝐻 𝑥
Desired properties:Ground energy of is small Verification circuit accepts a state with high probability is a yes instance
Key intermediate step: Design a Hamiltonian where each ground state encodes a quantum computation associated with the circuit, e.g., Feynman-Kitaev Hamiltonian has ground states
QMA Verification circuit for Hamiltonian
Proving QMA-hardness for ground energy problems
Wm-1Wm-2…W0¿𝜓 ⟩
¿0 ⟩⊗𝑛𝑎 𝐻 𝑥
Desired properties:Ground energy of is small Verification circuit accepts a state with high probability is a yes instance
Key intermediate step: Design a Hamiltonian where each ground state encodes a quantum computation associated with the circuit, e.g., Feynman-Kitaev Hamiltonian has ground states
QMA Verification circuit for Hamiltonian
In our case we show how to encode the history of an -qubit, -gate computation in the groundspace of the -particle Bose-Hubbard model on a graph with vertices
When the Hamiltonian is just the adjacency matrix of the graph. We use a variant of the Feynman-Kitaev circuit-to-Hamiltonian mapping which outputs a symmetric 0-1 matrix.
Encoding one qubit with one particle
H H HT (HT)† HT (HT)† H H
When the Hamiltonian is just the adjacency matrix of the graph. We use a variant of the Feynman-Kitaev circuit-to-Hamiltonian mapping which outputs a symmetric 0-1 matrix.
Example (building block for later on)
Encoding one qubit with one particle
Vertices are labeled(𝑧 , 𝑡 , 𝑗 ) 𝑧∈ {0,1 } 𝑡 , 𝑗∈ [8]
H H HT (HT)† HT (HT)† H H
Groundstates of the adjacency matrix:
When the Hamiltonian is just the adjacency matrix of the graph. We use a variant of the Feynman-Kitaev circuit-to-Hamiltonian mapping which outputs a symmetric 0-1 matrix.
Example (building block for later on)
Encoding one qubit with one particle
a specific state encodes the computation where the circuit is complex-conjugated
for
Vertices are labeled(𝑧 , 𝑡 , 𝑗 ) 𝑧∈ {0,1 } 𝑡 , 𝑗∈ [8]
More particles ()
𝐻𝐺(𝑡 ,𝑈 )=𝑡 ∑𝑖 , 𝑗 ∈𝑉
𝐴(𝐺)𝑖𝑗 𝑎𝑖† 𝑎 𝑗+𝑈 ∑𝑘∈𝑉
𝑛𝑘 (𝑛𝑘−1 )
0
= smallesteigenvalue of
More particles ()
𝐻𝐺(𝑡 ,𝑈 )=𝑡 ∑𝑖 , 𝑗 ∈𝑉
𝐴(𝐺)𝑖𝑗 𝑎𝑖† 𝑎 𝑗+𝑈 ∑𝑘∈𝑉
𝑛𝑘 (𝑛𝑘−1 )
0
If the ground energy is we say the groundspace is frustration-free. In this case the groundspace is the same for all>0!
= smallesteigenvalue of
More particles ()
We design a class of graphs where the frustration-free -particle ground states encode -qubit computations. These graphs are built out of multiple copies of
𝐻𝐺(𝑡 ,𝑈 )=𝑡 ∑𝑖 , 𝑗 ∈𝑉
𝐴(𝐺)𝑖𝑗 𝑎𝑖† 𝑎 𝑗+𝑈 ∑𝑘∈𝑉
𝑛𝑘 (𝑛𝑘−1 )
0
If the ground energy is we say the groundspace is frustration-free. In this case the groundspace is the same for all>0!
= smallesteigenvalue of
4096 vertex graphmade from32 copies of
Graphs for two-qubit gates
Two qubit gate A graph shaped like this
𝑈
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Two-particle frustration-free ground states have the form
1√2
|both particles on the ¿𝜙 ⟩+ 1√2 ¿𝑈 𝜙 ¿¿
Single-particle ground states encode a qubit and one out of four possible locations
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
1√2
|both particles on the ¿𝜙 ⟩+ 1√2 ¿𝑈 𝜙 ¿¿
Single-particle ground states encode a qubit and one out of four possible locations
Two-particle frustration-free ground states have the form
Two qubit gate A graph shaped like this
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Single-particle ground states encode a qubit and one out of four possible locations
1√2
|both particles on the ¿𝜙 ⟩+ 1√2 ¿𝑈 𝜙 ¿¿
Two qubit gate A graph shaped like this
Two-particle frustration-free ground states have the form
𝑈 Graphs for two-qubit gates
4096 vertex graphmade from32 copies of
Good news: there are two-particle ground states which encode computations
,𝑈 1𝜙,𝜙+¿ ,𝑈 1𝜙+¿,𝑈 1𝜙+¿ ,𝑈 1𝜙+¿ ,𝑈 2𝑈 1𝜙+¿
Constructing a graph from a verification circuit
𝑈 1 𝑈 2
Bad news: there are other two-particle ground states where the particles are in regions of the graph where they shouldn’t be.
To get rid of the bad states, we develop a method for enforcing “occupancy constraints”, i.e. penalizing certain configurations of the particles. To prove our results we establish spectral bounds without using perturbation theory.
Connect up the graphs for two-qubit gates to mimic the circuit, e.g.,
• Improvements to our construction? E.g., remove restriction to fixed particle number and/or consider simple graphs (no self loops).
• Complexity of other models of indistinguishable particles on graphs?– bosons or fermions with nearest-neighbor interactions– Attractive interactions– Negative hopping strength
• Complexity of other spin models on graphs?– Antiferromagnetic Heisenberg model– Models with only one type of 2-local term: for which matrices is the model
QMA-complete?
Open questions