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Quantum 3-SAT is QMA1-complete
David Gosset (Institute for Quantum Computing, University of Waterloo)
Daniel Nagaj (University of Vienna)
Long version: arXiv: 1302.0290
Short version : Proceedings of FOCS 2013
Quantum k-SAT (Bravyi 2006)Each clause is a k-local projector and is satisfied by a state if .
The amount that violates a clause is
Quantum k-SAT (Bravyi 2006)Each clause is a k-local projector and is satisfied by a state if .
The amount that violates a clause is
Quantum k-SATGiven k-local projectors {. We are promised that either
(YES) There is a state which satisfies for each
(NO) for all states
and asked to decide which is the case.
Quantum k-SAT (Bravyi 2006)Each clause is a k-local projector and is satisfied by a state if .
The amount that violates a clause is
Quantum k-SATGiven k-local projectors {. We are promised that either
(YES) There is a state which satisfies for each
(NO) for all states
and asked to decide which is the case.
Exactly satisfies eachclause
Quantum k-SAT (Bravyi 2006)Each clause is a k-local projector and is satisfied by a state if .
The amount that violates a clause is
Quantum k-SATGiven k-local projectors {. We are promised that either
(YES) There is a state which satisfies for each
(NO) for all states
and asked to decide which is the case.
Exactly satisfies eachclause
Total violation is at least 1. Can be obtained from by repeating each term
Quantum k-SAT (Bravyi 2006)Each clause is a k-local projector and is satisfied by a state if .
The amount that violates a clause is
Quantum k-SATGiven k-local projectors {. We are promised that either
(YES) There is a state which satisfies for each
(NO) for all states
and asked to decide which is the case.
Exactly satisfies eachclause
Total violation is at least 1. Can be obtained from by repeating each term
Classical k-SAT is the special case where all projectors are diagonal
Quantum k-SAT is a special case of k-local Hamiltonian where the Hamiltonian is frustration-free for yes instances
k
k-local Hamiltonian problem
Quantum k-SAT
Classical k-SAT
Yes instances are frustration-free
All constraints are diagonal
k
k-local Hamiltonian problem
Quantum k-SAT
Classical k-SAT
Yes instances are frustration-free
All constraints are diagonal
Complexity of quantum k-SAT
k=2
๐โฅ4
Contained in P
QMA1-complete
๐=๐๐=๐
4
[Bravyi 2006]
also follows from [Kitaev 99])
k
k-local Hamiltonian problem
Quantum k-SAT
Classical k-SAT
Yes instances are frustration-free
All constraints are diagonal
Complexity of quantum k-SAT
k=2
๐โฅ4
Contained in P
QMA1-complete
Contained in QMA1
NP-hard
๐=๐๐=๐
4
[Bravyi 2006]
also follows from [Kitaev 99])
k
k-local Hamiltonian problem
Quantum k-SAT
Classical k-SAT
Yes instances are frustration-free
All constraints are diagonal
Complexity of quantum k-SAT
We prove quantum 3-SAT is QMA1-hard (and therefore QMA1-complete).
k=2
๐โฅ4
Contained in P
QMA1-complete
Contained in QMA1
NP-hard
๐=๐๐=๐
4
[Bravyi 2006]
also follows from [Kitaev 99])
k
k-local Hamiltonian problem
Quantum k-SAT
Classical k-SAT
Yes instances are frustration-free
All constraints are diagonal
Complexity of quantum k-SAT
k=2
๐โฅ4
Contained in P
QMA1-complete
๐=๐๐โฅ๐
[Ambainis Kempe Sattath 2010][Arad Sattath 2013][Schwarz Cubitt Verstraete 2013]
Many authors have studied quantum SAT since Bravyiโs work
Quantum Lovรกsz Local Lemma
[Laumann Lรคuchli Moessner Scardicchio Sondhi 2010][Laumann Moessner Scardicchio Sondhi 2010][Bravyi Moore Russell 2010][Hsu Laumann Lรคuchli Moessner Sondhi 2013][Bardoscia Nagaj Scardicchio 2013]
Ensembles of randominstances of quantum k-SAT
[Eldar Regev 2008] Complexity of quantum 2-SAT with higher dimensional particles (qudits)
[Ji Wei Zeng 2011] Characterization of the groundspace of yes instances of quantum 2-SAT
[Sattath 2013] โAn almost sudden jump in quantum complexityโ
QMA1
If is a yes instance there exists (a witness) which is accepted with probability exactly 1.If is a no instance every state is accepted with probability at most
Wm-1Wm-2โฆW0
ยฟ๐ โฉยฟ0 โฉโ๐๐
QMA1 verification circuit
Because of the perfect completeness, the definition of QMA1 is gate-set dependent.It is not known whether or not QMA=QMA1; see
[Aaronson 2009] [Jordan, Kobayashi, Nagaj, Nishimura 2012][Kobayashi, Le Gall, Nishimura 2013] [Pereszlenyi 2013]
QMA1 is a one-sided error version of QMA. This is the relevant class becausequantum k-SAT is defined with one-sided error.
Bravyi proved quantum k-SAT is contained in QMA1 (verification circuit: choose one projector at random and measure it).
Bravyi proved quantum k-SAT is contained in QMA1 (verification circuit: choose one projector at random and measure it).
To prove QMA1-hardness of quantum 3-SAT we use a circuit-to-Hamiltonian mapping, i.e., we reduce from quantum circuit satisfiability.
If x is a yes instance there is an input state (witness) which makes the circuit output 1 with certainty. Ground energy of is zero.
If x is a no instance no input state makes the circuit output 1 with probability greater than Ground energy of is at least .
QMA1-hardness via circuit-to-Hamiltonian mapping
Wm-1Wm-2โฆW0ยฟ๐ โฉ
ยฟ0 โฉโ๐๐ ๐ป ๐ฅ=โ๐ฮ ๐
QMA1 Verification circuit for Quantum 3-SAT Hamiltonian
Wm-1Wm-2โฆW0ยฟ๐ โฉยฟ0 โฉโ๐๐
Hilbert space
QMA1 verification circuit (n qubits, m gates)
|๐ง โฉ|๐ก โฉ ๐งโ {0,1 }๐ , ๐กโ{0,1,2 ,โฆ,๐ }
Example part 1 [Kitaev 99]
Wm-1Wm-2โฆW0ยฟ๐ โฉยฟ0 โฉโ๐๐
Hilbert space
QMA1 verification circuit (n qubits, m gates)
|๐ง โฉ|๐ก โฉ ๐งโ {0,1 }๐ , ๐กโ{0,1,2 ,โฆ,๐ }
๐ป๐ก ,๐ก+1 (๐ ๐ก )=12 ยฟTransitionoperators
Example part 1 [Kitaev 99]
Wm-1Wm-2โฆW0ยฟ๐ โฉยฟ0 โฉโ๐๐
Hilbert space
Transitionoperators
QMA1 verification circuit (n qubits, m gates)
|๐ง โฉ|๐ก โฉ ๐งโ {0,1 }๐ , ๐กโ{0,1,2 ,โฆ,๐ }
๐ป๐ก ,๐ก+1 (๐ ๐ก )=12 ยฟ
๐ป ๐น๐๐ฆ๐๐๐๐=โ๐=1
๐๐|1 โฉ โจ1|๐โโจ0โฉโจ 0โจยฟ+โ
๐ก=0
๐โ1
๐ป๐ก ,๐ก+1(๐ ๐ก)+|0 โฉ โจ0|๐๐ข๐กโโจ๐โฉโจ๐โจยฟยฟHamiltonian
Example part 1 [Kitaev 99]
Wm-1Wm-2โฆW0ยฟ๐ โฉยฟ0 โฉโ๐๐
Hilbert space
Transitionoperators
QMA1 verification circuit (n qubits, m gates)
|๐ง โฉ|๐ก โฉ ๐งโ {0,1 }๐ , ๐กโ{0,1,2 ,โฆ,๐ }
๐ป๐ก ,๐ก+1 (๐ ๐ก )=12 ยฟ
Hamiltonian
1โ๐+1
(|๐ โฉ|0 โฉ+๐ 0|๐ โฉ|1 โฉ+๐ 1๐ 0|๐ โฉ|2 โฉ+โฆ+๐๐โ1๐๐โ2โฆ๐ 0โจ๐ โฉโจ๐โฉ)Nullspace consists of โhistory statesโ
๐ป ๐น๐๐ฆ๐๐๐๐=โ๐=1
๐๐|1 โฉ โจ1|๐โโจ0โฉโจ 0โจยฟ+โ
๐ก=0
๐โ1
๐ป๐ก ,๐ก+1(๐ ๐ก)+|0 โฉ โจ0|๐๐ข๐กโโจ๐โฉโจ๐โจยฟยฟ
Example part 1 [Kitaev 99]
Wm-1Wm-2โฆW0ยฟ๐ โฉยฟ0 โฉโ๐๐
Hilbert space
Transitionoperators
QMA1 verification circuit (n qubits, m gates)
|๐ง โฉ|๐ก โฉ ๐งโ {0,1 }๐ , ๐กโ{0,1,2 ,โฆ,๐ }
๐ป๐ก ,๐ก+1 (๐ ๐ก )=12 ยฟ
Hamiltonian
1โ๐+1
(|๐ โฉ|0 โฉ+๐ 0|๐ โฉ|1 โฉ+๐ 1๐ 0|๐ โฉ|2 โฉ+โฆ+๐๐โ1๐๐โ2โฆ๐ 0โจ๐ โฉโจ๐โฉ)Nullspace consists of โhistory statesโ
To have zero energy for the other two terms, we must have|๐ โฉ=|0 โฉ๐๐โจ๐ โฉA witness accepted with probability 1
๐ป ๐น๐๐ฆ๐๐๐๐=โ๐=1
๐๐|1 โฉ โจ1|๐โโจ0โฉโจ 0โจยฟ+โ
๐ก=0
๐โ1
๐ป๐ก ,๐ก+1(๐ ๐ก)+|0 โฉ โจ0|๐๐ข๐กโโจ๐โฉโจ๐โจยฟยฟ
Example part 1 [Kitaev 99]
Wm-1Wm-2โฆW0ยฟ๐ โฉยฟ0 โฉโ๐๐
Hilbert space
QMA1 verification circuit (n qubits, m gates)
|๐ง โฉ|๐ก โฉ ๐งโ {0,1 }๐ , ๐กโ{0,1,2 ,โฆ,๐ }
๐ป๐ก ,๐ก+1 (๐ ๐ก )=12 ยฟ
has a zero energy ground state if and only if the QMA1 verification circuit accepts a witness with probability 1. However, itโs not local.
Example part 1 [Kitaev 99]
Kitaev used a clock construction to convert it to a local Hamiltonianโฆ
Transitionoperators
Hamiltonian ๐ป ๐น๐๐ฆ๐๐๐๐=โ๐=1
๐๐|1 โฉ โจ1|๐โโจ0โฉโจ 0โจยฟ+โ
๐ก=0
๐โ1
๐ป๐ก ,๐ก+1(๐ ๐ก)+|0 โฉ โจ0|๐๐ข๐กโโจ๐โฉโจ๐โจยฟยฟ
Hilbert space HcompโHclock
m qubitsn qubits
Example part 2: Clock construction [Kitaev 99]
๐ป ๐พ๐๐ก๐๐๐ฃ=1โโ๐=1
๐โ1
|01 โฉ โจ 01ยฟ๐ ,๐+1+๐ป๐ ๐๐
Hilbert space HcompโHclock
m qubitsn qubits
Hamiltonian
Example part 2: Clock construction [Kitaev 99]
A sum of 5-local projectors
๐ป ๐พ๐๐ก๐๐๐ฃ=1โโ๐=1
๐โ1
|01 โฉ โจ 01ยฟ๐ ,๐+1+๐ป๐ ๐๐
Nullspace spanned by
Hilbert space HcompโHclock
m qubitsn qubits
|t โฉ๐ข=|111โฆ1000โฆ0 โฉ ,t=0 ,โฆ,m
๐ก ๐โ๐ก
Hamiltonian
Example part 2: Clock construction [Kitaev 99]
A sum of 5-local projectors
๐ป ๐พ๐๐ก๐๐๐ฃ=1โโ๐=1
๐โ1
|01 โฉ โจ 01ยฟ๐ ,๐+1+๐ป๐ ๐๐
๐ป ๐ ๐๐|HcompโSclock=๐ป ๐น๐๐ฆ๐๐๐๐
Nullspace spanned by
is designed so that
This implies has the same nullspace as
Hilbert space HcompโHclock
m qubitsn qubits
|t โฉ๐ข=|111โฆ1000โฆ0 โฉ ,t=0 ,โฆ,m
๐ก ๐โ๐ก
Hamiltonian
Example part 2: Clock construction [Kitaev 99]
A sum of 5-local projectors
h๐ก ,๐ก+1 (๐ ๐ก ) |HcompโS clock=๐ป๐ก ,๐ก+1(๐ ๐ก)
๐0|Sclock=|0 โฉ โจ 0โจยฟ
๐๐|Sclock=|๐ โฉ โจ๐โจยฟ
This is achieved โterm by termโ, by exhibiting projectors (acting on ) and projectors acting on such that
Example part 2: Clock construction [Kitaev 99]
h๐ก ,๐ก+1 (๐ ๐ก ) |HcompโS clock=๐ป๐ก ,๐ก+1(๐ ๐ก)
๐0|Sclock=|0 โฉ โจ 0โจยฟ
๐๐|Sclock=|๐ โฉ โจ๐โจยฟ
This is achieved โterm by termโ, by exhibiting projectors (acting on ) and projectors acting on such that
Example part 2: Clock construction [Kitaev 99]
๐ป ๐พ๐๐ก๐๐๐ฃ=1โโ๐=1
๐โ1
|01 โฉ โจ 01ยฟ๐ ,๐+1+โ๐=1
๐๐|1 โฉ โจ1|๐โ๐0+โ
๐ก=0
๐โ1
h๐ก , ๐ก+1(๐ ๐ก)+|0 โฉ โจ0|๐๐ข๐กโ๐๐
h๐ก ,๐ก+1 (๐ ๐ก ) |HcompโS clock=๐ป๐ก ,๐ก+1(๐ ๐ก)
๐0|Sclock=|0 โฉ โจ 0โจยฟ
๐๐|Sclock=|๐ โฉ โจ๐โจยฟ
This is achieved โterm by termโ, by exhibiting projectors (acting on ) and projectors acting on such that
Example part 2: Clock construction [Kitaev 99]
๐ป ๐พ๐๐ก๐๐๐ฃ=1โโ๐=1
๐โ1
|01 โฉ โจ 01ยฟ๐ ,๐+1+โ๐=1
๐๐|1 โฉ โจ1|๐โ๐0+โ
๐ก=0
๐โ1
h๐ก , ๐ก+1(๐ ๐ก)+|0 โฉ โจ0|๐๐ข๐กโ๐๐
A -local projector if is j-local
1-local projectors
h๐ก ,๐ก+1 (๐ ๐ก ) |HcompโS clock=๐ป๐ก ,๐ก+1(๐ ๐ก)
๐0|Sclock=|0 โฉ โจ 0โจยฟ
๐๐|Sclock=|๐ โฉ โจ๐โจยฟ
This is achieved โterm by termโ, by exhibiting projectors (acting on ) and projectors acting on such that
Kitaevโs Hamiltonian is a sum of k-local projectors with for circuits made from 1- and 2-qubit gates.
Kitaevโs construction can be used to prove that quantum 5-SAT is QMA1-hard.
Example part 2: Clock construction [Kitaev 99]
๐ป ๐พ๐๐ก๐๐๐ฃ=1โโ๐=1
๐โ1
|01 โฉ โจ 01ยฟ๐ ,๐+1+โ๐=1
๐๐|1 โฉ โจ1|๐โ๐1+โ
๐ก=0
๐ โ1
h๐ก ,๐ก+1(๐ ๐ก)+|0 โฉ โจ0|๐๐ข๐กโ๐๐
A -local projector if is j-local
1-local projectors
The first ingredient in our QMA1-hardness proof is a new clock construction (with different locality from Kitaevโs)โฆ
Properties of the new clock construction
๐ป๐๐๐๐๐๐
.
Hc lockSum of 3-local projectors Hamiltonian acting on
7N-3 qubits
Nullspace
ClockHamiltonian
Properties of the new clock construction
๐ป๐๐๐๐๐๐
.
Hc lockSum of 3-local projectors Hamiltonian acting on
HcompโHclock
7N-3 qubits
Nullspace
ClockHamiltonian
Transitionoperators
act on
A -local projector if U is j-local
Properties of the new clock construction
๐ป๐๐๐๐๐๐
.
Hc lockSum of 3-local projectors Hamiltonian acting on
HcompโHclock
7N-3 qubits
Nullspace
ClockHamiltonian
Transitionoperators
act on
Greater than/Less than operators ๐ถโค ๐
๐ถโค ๐|Sclock = โ1โค ๐<๐
|๐ถ ๐ โฉโจ ๐ถ ๐โจ+ยฟ12|๐ถ๐ โฉโจ ๐ถ๐โจยฟยฟ ๐ถโฅ ๐|Sclock =
12|๐ถ๐ โฉโจ ๐ถ๐โจ+ โ
๐< ๐ โค๐|๐ถ ๐ โฉโจ ๐ถ ๐โจยฟยฟ
act on Hclock
A -local projector if U is j-local
1-local projectors
3-local 2-local 4-local 2-local
Like Kitaevโs clock construction, ours could be used to emulate Feynmanโs Hamiltonian
This isnโt good enough for our purposesโit only shows that quantum 4-SAT is QMA1-hard (already known).
Instead, we use our clock construction in a different wayโฆ
1โ๐ป๐๐๐๐๐๐+ 1 +โ
๐=1
๐๐|1 โฉโจ 1โจ๐โ๐ถโค 1+โ
๐กh๐ก ,๐ก+1 (๐ ๐ก )+|0 โฉ โจ 0โจ๐๐ข๐กโ๐ถโฅ๐+1
Two clock registers
We map the verification circuit to a Hamiltonian acting on a Hilbert space with one n-qubit computational register and two clock registers:
2D grid of zero energy clock states
1โ๐ป๐๐๐๐๐๐ โ1+1โ1โ๐ป๐๐๐๐๐
๐
โInitialโ โFinalโ
Two clock registers
We map the verification circuit to a Hamiltonian acting on a Hilbert space with one n-qubit computational register and two clock registers:
1โ๐ป๐๐๐๐๐๐ โ1+1โ1โ๐ป๐๐๐๐๐
๐ +๐ป ๐๐๐๐
Every zero energy groundstate encodes the history ofa computation
Two clock registers
We map the verification circuit to a Hamiltonian acting on a Hilbert space with one n-qubit computational register and two clock registers:
for 1-local U
1โ๐ป๐๐๐๐๐๐ โ1+1โ1โ๐ป๐๐๐๐๐
๐ +๐ป ๐๐๐๐
is built out of 3-local projectors such as
h ๐ ,๐+1 (๐ )โ1
1โ๐ถโฅ๐โ๐ถโค ๐
|0 โฉ โจ 0โจ๐โh ๐ ,๐+1โ11โh๐ , ๐+1โ๐ถโค ๐
Every zero energy groundstate encodes the history ofa computation
(writing )
Two clock registers
We map the verification circuit to a Hamiltonian acting on a Hilbert space with one n-qubit computational register and two clock registers:
1โ๐ป๐๐๐๐๐๐ โ1+1โ1โ๐ป๐๐๐๐๐
๐ +๐ป ๐๐๐๐+โ๐=1
๐๐|1 โฉ โจ 1โจ๐โ๐ถโค 1โ๐ถโค 1+|0 โฉ โจ 0โจ๐๐ข๐กโ๐ถโฅ๐โ๐ถโฅ๐
Enforce initialization of ancillasand correct output of circuit
for 1-local U
is built out of 3-local projectors such as
h ๐ ,๐+1 (๐ )
1โ๐ถโฅ๐โ๐ถโค ๐
|0 โฉ โจ 0โจ๐โh ๐ ,๐+1โ11โh๐ , ๐+1โ๐ถโค ๐ (writing )
Two clock registers
We map the verification circuit to a Hamiltonian acting on a Hilbert space with one n-qubit computational register and two clock registers:
1โ๐ป๐๐๐๐๐๐ โ1+1โ1โ๐ป๐๐๐๐๐
๐ +๐ป ๐๐๐๐+โ๐=1
๐๐|1 โฉ โจ 1โจ๐โ๐ถโค 1โ๐ถโค 1+|0 โฉ โจ 0โจ๐๐ข๐กโ๐ถโฅ๐โ๐ถโฅ๐
Enforce initialization of ancillasand correct output of circuit
I will now show you how to construct for the case where the verification circuit is a specific two-qubit gate (warning: gadgetry ahead)โฆ
for 1-local U
is built out of 3-local projectors such as
h ๐ ,๐+1 (๐ )
1โ๐ถโฅ๐โ๐ถโค ๐
|0 โฉ โจ 0โจ๐โh ๐ ,๐+1โ11โh๐ , ๐+1โ๐ถโค ๐ (writing )
Zero energy ground states
1 2 3 4 5 6 7 8 91
2
3
4
5
6
7
8
9
1โ1โ๐ป ๐๐๐๐๐9 +1โ๐ป๐๐๐๐๐
9 โ1
Two clock registers: Example
Zero energy ground states is a vertex in the above graph
1 2 3 4 5 6 7 8 91
2
3
4
5
6
7
8
9
Two clock registers: Example
1โ1โ๐ป ๐๐๐๐๐9 +1โ๐ป๐๐๐๐๐
9 โ1+1โ๐ถโฅ 3โ๐ถโค1
Zero energy ground states
1 2 3 4 5 6 7 8 91
2
3
4
5
6
7
8
9
is a vertex in the above graph
Two clock registers: Example
+1โ๐ถโค 1โ๐ถโฅ 3+1โ๐ถโฅ 3โ๐ถโค11โ1โ๐ป ๐๐๐๐๐9 +1โ๐ป๐๐๐๐๐
9 โ1
1 2 3 4 5 6 7 8 91
2
3
4
5
6
7
8
9
|๐ โฉ๐๐|ฮ โฉ=|๐ โฉ๐๐ โ๐ , ๐โ ฮ
|๐ถ๐ โฉโจ๐ถ ๐โฉ where is a connected component of the graph
Zero energy ground states
Two clock registers: Example
+1โh12โ๐ถโค2+1โ๐ถโค 1โ๐ถโฅ 3+1โ๐ถโฅ 3โ๐ถโค11โ1โ๐ป ๐๐๐๐๐9 +1โ๐ป๐๐๐๐๐
9 โ1
1 2 3 4 5 6 7 8 91
2
3
4
5
6
7
8
9
|๐ โฉ๐๐|ฮ โฉ=|๐ โฉ๐๐ โ๐ , ๐โ ฮ
|๐ถ๐ โฉโจ๐ถ ๐โฉ where is a connected component of the graph
Zero energy ground states
Two clock registers: Example
+1โh12โ๐ถโค2+1โ๐ถโค 1โ๐ถโฅ 3+1โ๐ถโฅ 3โ๐ถโค11โ1โ๐ป ๐๐๐๐๐9 +1โ๐ป๐๐๐๐๐
9 โ1
Continuing in this way,we can design a Hamiltonian with ground states described by a more complicated graphโฆ
Built out of terms likeh ๐ ,๐+1โ๐ถโค๐
๐ถโค๐โ h๐ ,๐+1๐ถโฅ ๐โ๐ถโค ๐
|๐ โฉ๐๐|ฮ โฉ=|๐ โฉ๐๐ โ๐ , ๐โ ฮ
|๐ถ๐ โฉโจ๐ถ ๐โฉ where is a connected component of the graph
1 2 3 4 5 6 7 8 91
2
3
4
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7
8
9
Zero energy ground states
Two clock registers: Example
Commutes with
1 2 3 4 5 6 7 8 91
2
3
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8
9
Zero energy ground states
+ยฟ 0โฉโจ 0โจ๐โ (h34+h67 )โ1+ยฟ1โฉโจ 1โจ๐โ1โ (h34+h67 )
Two clock registers: Example
sector1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 91
2
3
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9
sector
|0โฉ๐โจ๐ โฉ๐|ฮ โฉ is a connected component
|1โฉ๐โจ๐ โฉ๐|ฮ โฉ is a connected component
Zero energy ground states Zero energy ground states
+ยฟ 0โฉโจ 0โจ๐โ (h34+h67 )โ1+ยฟ1โฉโจ 1โจ๐โ1โ (h34+h67 )
Two clock registers: Example
sector1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 91
2
3
4
5
6
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9
sector
|0โฉ๐โจ๐ โฉ๐|ฮ โฉ is a connected component
|1โฉ๐โจ๐ โฉ๐|ฮ โฉ is a connected component
Zero energy ground states Zero energy ground states
+ยฟ 0โฉโจ 0โจ๐โ (h34+h67 )โ1+ยฟ1โฉโจ 1โจ๐โ1โ (h34+h67 )
Two clock registers: Example
sector1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
|0 โฉ๐|๐ โฉ๐|โฉ
1 2 3 4 5 6 7 8 91
2
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9
sector
|0 โฉ๐|๐ โฉ๐|โฉ |1 โฉ๐|๐ โฉ๐|โฉ |1 โฉ๐|๐ โฉ๐|โฉ+ others + others
Zero energy ground states Zero energy ground states
+ยฟ 0โฉโจ 0โจ๐โ (h34+h67 )โ1+ยฟ1โฉโจ 1โจ๐โ1โ (h34+h67 )
Two clock registers: Example
sector1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 91
2
3
4
5
6
7
8
9
sector
๐ ๐
๐๐ ๐๐
๐ ๐
Zero energy ground states Zero energy ground states
+h45 (๐๐)โ1+h45ยฟ
Acts on first clock register and qubit b
Acts on second clock register and qubit b
[๐ ,๐ ]โ 0
+ยฟ 0โฉโจ 0โจ๐โ (h34+h67 )โ1+ยฟ1โฉโจ 1โจ๐โ1โ (h34+h67 )
Two clock registers: Example
sector1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
1 2 3 4 5 6 7 8 91
2
3
4
5
6
7
8
9
sector
+h45 (๐๐)โ1+h45ยฟ
๐ ๐
๐๐ ๐๐
๐ ๐
Acts on first clock register and qubit b
Acts on second clock register and qubit b
[๐ ,๐ ]โ 0
Zero energy ground states Zero energy ground states
+ยฟ 0โฉโจ 0โจ๐โ (h34+h67 )โ1+ยฟ1โฉโจ 1โจ๐โ1โ (h34+h67 )
Two clock registers: Example
The point is that every zero energy ground state encodes the history of a two-qubit computation
|๐ โฉ๐๐|๐ถ1 โฉ|๐ถ1 โฉ+โฆ+๐|๐ โฉ๐๐โจ๐ถ9 โฉโจ๐ถ9โฉ
where ๐=|0 โฉ โจ 0โจโ๐๐+ยฟ1โฉโจ 1โจโ๐๐
(An entangling two-qubit unitary for suitably chosen )
+h45 (๐๐)โ1+h45ยฟ
Acts on first clock register and qubit b
Acts on second clock register and qubit b
[๐ ,๐ ]โ 0
+ยฟ 0โฉโจ 0โจ๐โ (h34+h67 )โ1+ยฟ1โฉโจ 1โจ๐โ1โ (h34+h67 )
Two clock registers: Example
This was achieved without using the transition operator
Remarks and open questions
โข Are there simpler โclause-by-clauseโ reductions for quantum k-SAT? In the classical case there is a clause-by-clause way to map a (k+1)-SAT instance to a k-SAT instance, for .
โข Other applications for our new clock construction?
โข โFrustration-freeโ gadgetry has the advantage over perturbation theory methods that one can avoid large (system size dependent) terms in the Hamiltonian.