Behavior of O(log n) Local Commuting Hamiltonians

(1)

Behavior of

O

(

log

n

)

local commuting hamiltonians

Thesis by

Jenish C. Mehta

In Partial Fulfillment of the Requirements for the degree of

Master of Science

CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California

(2)

c

2016

(3)

Acknowledgements

(4)

Abstract

We study the variant of thek-local hamiltonian problem which is a natural generalization of

k-CSPs, in which the hamiltonian terms all commute. More specifically, we consider a hamil-tonianH=_∑_iHiovernqubits, where eachHiacts non-trivially onO(logn)qubits and all the termsHicommute, and show the following

-1. We show that a specific case ofO(logn)local commuting hamiltonians over the hyper-cube is inNP, using the Bravyi-Vyalyi Structure theorem from [7].

2. We give a simple proof of a generalized area law for commuting hamiltonians (which seems to be a folklore result) in all dimensions, and deduce the case forO(logn)local commuting hamiltonians.

3. We show that traversing the ground space ofO(logn)local commuting hamiltonians is

QCMAcomplete.

(5)

or PSPACE (and whether complete for it). The quantum analogue of the problem - traversing the ground space of hamiltonians was studied in [10]. Once again, the problem is defined as follows - Given a local hamiltonian H on nqubits and two unitaries that efficiently prepare the starting and final statesψandφwith energy close to the ground space of H, is there a sequence

of poly(n)local unitaries such that the ground states of H can be traversed, i.e., we can move from ψto a state close toφ while all intermediate states have energy close to the ground space

energy? It was shown in [10], that traversing the ground space of 5-local hamiltonians, (when the number of intermediate unitaries are polynomial in the number of qubits) isQCMAcomplete. This was indeed the first problem shown to be complete for the classQCMA. We consider the commuting variant of the problem. But unlike the previous two observations, we indeed show that traversing the ground space ofO(logn) local commuting hamiltonians is QCMAcomplete. We present a novel construction that helps us use the tools from [10] for the commuting case after certain modifications. Thus, in this case, it indeed seems the case that traversing the ground space ofO(logn)local commuting hamiltonians is just as hard as traversing the ground space of 5-local hamiltonians.

Thus, we seem to have indications in both directions about the hardness of theO(logn)-CLHP. The first two obervations seem to indicate that the problem is easier thanQMA, while the last result seems to indicate that the problem might indeed beQMA-hard. One thing to note is that in case of the first results, the hamiltonians were geometrically local, while in our ground space traversal construction, the hamiltonians are highlynon-local. Is it the case that forO(logn)local commuting hamiltonians, geometrically local hamiltonians are easier than the ones that are not-geometrically local? Such a fact would indeed be surprising, since it does not hold for thek-LHP. We state other such problems towards the end.

In section 2, we review certain background definitions and results. In section 3, we give an exposition of the Structure Theorem as in [7], and some of its applications. In section 4, we give a proof of a generalized area law for commuting hamiltonians in all dimensions. In section 5, we show that traversing the ground space ofO(logn)local commuting hamiltonians is QCMA

(9)

(10)

there is some deterministic polynomial time machineMsuch that

x∈ L ⇒ E_r←U_poly

(|x|)

(M(x,y,r))≥ 2

3

x 6∈L ⇒ _E_r←U

poly(|x|)

(M(x,y,r))≤ 1

3

A language (subset of all set of strings over some alphabet) Lis in the complexity classNPif there is some deterministic polynomial time machineMsuch that

x∈ L ⇒ ∃y,|y| ≤poly(|x|),M(x,y) =1

x6∈L ⇒ ∀y,|y| ≤poly(|x|),M(x,y) =0

A language (subset of all set of strings over some alphabet) Lis in the complexity classMAif there is some deterministic polynomial time machineMsuch that

x∈ L ⇒ ∃y,|y| ≤poly(|x|),Er←U_poly

(|x|)

(M(x,y,r))≥ 2

3

x 6∈L ⇒ ∀y,|y| ≤poly(|x|),Er←U

poly(|x|)

(M(x,y,r))≤ 1

3

A language L is in the complexity class QCMA if there is some quantum polynomial time machineMsuch that

x ∈L ⇒ ∃y,|y| ≤poly(|x|),E(M(x,y))≥ 2

3

x6∈L ⇒ ∀y,|y| ≤poly(|x|),E(M(x,y))≤ 1

3

A language L is in the complexity classQMAif there is some quantum polynomial time ma-chineMsuch that

x∈ L ⇒ ∃v∈_C2poly(|x|)_,_E₍_M₍_x_,_v₎₎_≥ 2

3

x 6∈L ⇒ ∀v∈_C2poly(|x|)_,_E₍_M₍_x_,_v₎₎_≤ 1

3

The constants 2₃and1₃ are arbitrary and can be modified/amplified for the probabilistic classes above to (1−2−poly(n), 2−poly(n)). It is straightforward to show: P ⊆ NP ⊆ MA ⊆ QCMA ⊆

(11)

(12)

(13)

(14)

(15)

andDi_j0 are orthogonal. LetCi_j = Ci,j andAi⊗Di_j =Ri,j. Note that all the sets of operators except

Ci,j’s are orthogonal.

Extend the set of operatorsQk’s to a basis, and letRi,j =∑lαlQl. Then

H =

∑

i,j

Ci,j⊗Ri,j =

∑

i,j,l

αlCi,j⊗Ql =

∑

l

(αl

∑

i,j

Ci,j)⊗Ql =

∑

k

Bk⊗Qk

Now note that since the Qi’s are orthogonal and hence linearly independent, we get that Bk =

αk∑i,jCi,j. Since theBk’s are a linear combination of theCi,j’s, we get that the algebra generated by both are the same.

(16)

4 Area Laws

In this section, we change our focus, and consider area laws. We first review the main ideas in the combinatorial proof of [5] for showing an area law for Hamiltonians on a 1D chain. We then give a simple proof of a generalized area law for commuting hamiltonians in all dimensions.

4.1 Review of the combinatorial proof of the area law in 1D

The setup in [5] is as follows. We are given a local hamiltonianH =_∑n_i=₁H_ionnqudits (dimension

d) on a line. The assumptions are as follows:

1. Each Hi is 2-local (eachHi ∈(Cd)2) and an orthogonal projection, i.e. H2_i = Hi = Hi∗ 2. The hamiltonianHis frustration free, i.e.λmin(H) =0

3. There is a unique ground statevsuch thatHv=0 4. The smallest non-zero eigenvalue of His at leaste

Assumptions 1 and 3 are mild, in the sense that they can be achieved by scaling the hamiltonian terms appropriately (after assuming 2). Assumption 2 is strong, though in a later work [4], it was shown that the same ideas can be made to work for frustrated systems. Assumption 4 is strong too, since it requireseto be at least aconstantfor the proof to work. The bound on the entanglement

entropy of the ground state across the cut in [4] scales as log(1_e), and becomes weaker if one considerseto be inverse polynomial inn, though it is still stronger than a naive volume law.

We now state some definitions.

Definition 7. (Schmidt Rank) Given a system ofnqudits, letS ⊆ [n]be some set of qubits. Then for any vector (state)v, the schmidt rank ofv, denoted SRS(v)is the minimum integerDsuch that

v= ∑_i=D₁λiui⊗wi. For the minimumD, the set{ui}are orthonormal vectors overSand{wi}are orthonormal vectors overS. Further, theλi’s are in non-decreasing order, and strictly greater than 0. Similarly, given any operatorK, the schmidt rank ofK, denoted SRS(K)is the minimum integer

Dsuch thatK=∑_i=D₁Pi⊗Qi, where for the minimumD, the{Pi}and{Qi}are orthogonal (with respect to the trace norm) operators acting on the sets of quditsSandSrespectively.

We state a few simple properties of schmidt rank that will be useful for us.

Lemma 8. (i) SRS(u+v)≤SRS(u) +SRS(v)

(ii) for c >0, SRS(cu) =SRS(u)

(iii) for any two operators A and B, SRS(AB)≤SRS(A)SRS(B)

The proof of the above lemma follows trivially from the definitions.

One main tool used in [4] is the use of Approximate Ground State Projectors (AGSPs), which we define next.

Definition 9. (AGSP) LetH = ∑_i=n ₁Hi be a local hamiltonian on a chain as described above, and let(i,i+1)be the cut on the chain. An operatorKis a(D,γ)-AGSP if,

1. It preserves the ground space, i.e. if Hv=0, thenKv=v

2. It shrinks (and is invariant over) the space orthogonal to the ground space by a factorγ, i.e.,

for allworthogonal to the ground space,Kwis orthogonal to the ground space, andkKwk ≤λkwk

3. It has bond dimensionD, i.e.,SR≤i(K)≤D.

(17)

(18)

(19)

Remark13. The key thing to note in the above lemma is that the entanglement entropy is a function only of the number of hamiltonians acting across the cut. Further, this holds foranyconfiguration of hamiltonians, not necessary hamiltonians that are geometrically local, which is surprising, since it was shown in [3] that for general hamiltonians (not necessarily commuting) that are not geo-metrically local, there exist states with very high entanglement entropy across the cut. The above lemma is essentially the area law for commuting hamiltonians.

Corollary 14. Consider a L dimensional hypergrid of side length n consisting of nLqudits each of dimen-sion d. Let H = ∑n_i=L₁Hi be a system of frustration free commuting2L-local hamiltonians with a unique

ground state v, where each Hi acts over a hypercube of2Lqudits in a natural way. Let A be any contiguous

region of qudits. Then the number of hamiltonians acting from A to A is exactly|∂A|(where|∂A|denotes

the boundary of A). Thus, from the lemma above, S(v)≤ 2L|∂A|logd, which for constant L and d gives

S(v)≤ O(|∂A|). Further, note that if the hamiltonians have locality that is logarithm of the total number

of qubits, then2L= Llogn, and we would have (assuming d constant),

S(v)≤O(Llogn|∂A|)

In general for n qudits (constant d) and Hamiltonians with locality O(logn),

S(v)≤O(logn|∂A|)

which is much stronger than volume law, and is infact an area law for O(logn) local commuting hamiltonians.

(20)

(21)

(22)

(23)

(24)

P10=|111ih000|,

P+= 1₂(|000i+|111i)(h000|+h111|),

P−= 1₂(|000i − |111i)(h000| − h111|)

Lemma 22. (Small Projection Lemma) Let S and T be 2-orthogonal subspaces spanned by I⊗I⊗ |000i

and I⊗I⊗ |111irespectively. Let v0 = u⊗ |000iwhere u spans the first two registers. Let U1to Umbe

any sequence of 2-local unitaries, and let vi =Uivi−1for1≤i≤m. Further, assume we have that for every i,kI⊗I⊗Πvik ≤γ. Then, for every i,k(I⊗I⊗P1)vik ≤iγandk(I⊗I⊗P0)vik ≥1−(i+1)γ

Proof. First, note that since each Ui is a 2-local unitary, for any vector v, (I⊗ I⊗P1)Ui(I⊗I ⊗

P0)v=0 and(I⊗I⊗P0)Ui(I⊗I⊗P1)v=0. We now inductively show that for everyi,k(I⊗I⊗ P1)vik ≤iγ. It is trivially true fori=0. Let it be true fori−1. Then fori,

k(I⊗I⊗P1)vik = k(I⊗I⊗P1)Uivi−1k

= k(I⊗I⊗P1)Ui((I⊗I⊗P0) + (I⊗I⊗P1) + (I⊗I⊗Π))vi−1k

≤ k(I⊗I⊗P1)Ui(I⊗I⊗Π)vi−1k+k(I⊗I⊗P1)Ui(I⊗I⊗P1)vi−1k

≤ k(I⊗I⊗Π)vi−1k+k(I⊗I⊗P1)vi−1k

≤ γ+ (i−1)γ

= iγ

The second line used the fact that I = (I⊗I⊗P0) + (I⊗I⊗P1) + (I⊗I⊗Π), the third line

follows from triangle inequality and the fact that (I ⊗I⊗P1)Ui(I⊗I⊗P0)v = 0 sinceUi’s are 2-local and the subspaces 3 orthogonal, the fourth line follows from the fact thatkPk ≤1 for any orthogonal projectionP, and the fifth line uses the given condition and the inductive hypothesis.

Further,

k(I⊗I⊗P0)vik = kvi−(I⊗I⊗P1)vi−(I⊗I⊗Π)vik

≥ kvik − k(I⊗I⊗P1)vik − k(I⊗I⊗Π)vik

≥ 1−iγ−γ

= 1−(i+1)γ

≥ 1−(m+1)γ

Again, the second line uses the triangle inequality, and the third line uses the above proven claim and the given condition.

We are now ready to prove a modified version of the Traversal Lemma from [10] that we will require in the next section.

Lemma 23. (Modified Traversal Lemma) LetΠS= I⊗P0⊗P0and S be the +1 eigenspace ofΠS,ΠT =

I⊗P1⊗P0and T be the +1 eigenspace ofΠT. Define P= I⊗Π⊗P0, Q= I⊗(P0+P1)⊗P0. Let v0∈ S and w ∈ T, andkw−vmk ≤ e. If U1, ...,Um is a sequence of 2-local unitaries that map v0 to vm, with

vi =Uivi−1andkI⊗I⊗Πvik ≤γfor1≤i≤m, then there exists an i such thatkPvik2 ≥ 1₂ 1−₂_m2e

2

, providedγ≤ ₄_(m+1 ₁₎ 1−₂_m2e2.

Proof. We prove by contradiction. Let δ=1₂ 1−₂_m2e2. Assume that for all 1 ≤ i ≤ m, kPvik2 < δ.

(25)

(26)

5.4.1 Construction of commuting local hamiltonians

Let Aand Bbe two hamiltonians, not necessarily local, that act on n qubits, andkAk ≤ 1 and

kBk ≤1. Further, we have the following promise - Either there existsvsuch thathv,(A+B)vi ≤α,

or for allv,hv,(A+B)vi ≥ β. We shall now construct commuting Hamiltonians, traversing whose

ground space would help us comment on the ground state energy ofA+B.

Define the three new Hamiltonians as follows, acting on three registers, first of nqubits and the remaining two of 3 qubits each:

HA= A⊗Π⊗P+

HB = B⊗Π⊗P−

G= I⊗I⊗_Π

The initial and final states will bev0 = |0i ⊗ |000i ⊗ |000iandw = |0i ⊗ |111i ⊗ |000i. The

main idea in defining the hamiltonians as above is that because they are in a superposition on

|000iand|111i, the starting state on the third register does not make the energy on the states 0. Further, since you’ll always stay close to the|000istate as shown in lemma A above,HAandHB will always have high energy due to the third register, unless it is cancelled by the second or first register. The reason we need the third register is to make the three hamiltonians above commute. Note that HAHB = 0, and since the hamiltonian terms on the third register are all orthogonal projections (hermitian), they commute simply as HAG = GHA andHBG = GHB. Moreover, we cannot use less than 3 qubits in the third register, since we do not want any unitary to go from

|000i to |111i in 1 step on the third register, in which case it can traverse the ground space of

HA+HB+Gwithout traversing the ground space of A+B.

We now proceed to show that the construction above is both complete and sound.

5.4.2 Completeness

Assume there is some classical v such that hv,(A+B)vi ≤ α, then the following sequence of

unitaries help traverse the ground space ofHA+HB+G. The starting state isv0 = |0i ⊗ |000i ⊗

|000i.

1. Prepare the classical stringvin the first register using PauliXgates

2. Flip the bits of the second register from 000 to 111, again using PauliXgates. 3. Unprepare the stringvback to the all zeroes string in the first register.

Note that during the first and the third stage, we will remain in the ground state ofHAandHB by the virtue of the state in the second register, and in the ground state ofGdue to the state in the third register.

(27)

hv|ha|h000|(HA+HB+G)|vi|ai|000i = hv|ha|h000|(HA+HB)|vi|ai|000i +hv|ha|h000|G|vi|ai|000i

= hv|ha|h000|(A⊗Π⊗P++B⊗Π⊗P−)|vi|ai|000i

+hv|ha|h000|I⊗I⊗_Π|vi|ai|000i

= hv|A|viha|Π|aih000|P+|000i +hv|B|viha|Π|aih000|P−|000i+0

= 1

2hv|A|vi+ 1

2hv|B|vi

= 1

2hv|A+B|vi

≤ 1

2α

Thus, at any stage, the energy of any state is at most 1₂α, and thus we always remain close to

the ground space.

5.4.3 Soundness

To show the soundness, assume that for all classicalv,hv,(A+B)vi ≥β. Assume some sequence

of 2 qubit unitaries maps v0 to w and kvm −wk ≤ e for some e that will be set to a constant.

Further, assume thathvi,Gvii ≤γ2, because if not, then the energy will be atleastγ2. Thus, note

that for everyi, we have thatkvi,I⊗I⊗Πvik ≤γ.

Hence, applying the Modified Traversal Lemma, we would have that there is someifor which

hvi,I⊗Π⊗P0vii ≥δ. Then we have that

hvi,(HA+HB+G)vii ≥ hvi,(HA+HB)vii=hvi,(A⊗Π⊗P++B⊗Π⊗P−)vii

= hvi,(A+B)⊗Π⊗P0vii+E

≥ βδ+E

Now we bound the error term. Rewritevi = (I⊗I ⊗P0)vi+ (I⊗I⊗P1)vi+ (I⊗I⊗Π)vi. Thus,

E = hvi,(A−B)⊗Π⊗P01vii+hvi,(A−B)⊗Π⊗P10vii+hvi,(A+B)⊗Π⊗P1vii

= h(I⊗I⊗P0)vi,(A−B)⊗Π⊗P01P1vii+h(I⊗I⊗P1)vi,(A−B)⊗Π⊗P10P0vii

+h(I⊗I⊗P1)vi,(A+B)⊗Π⊗P1vii

(28)

(29)

(30)

|000iwhich can be easily prepared by polynomial sized circuits since |φican be prepared by a

polynomial sized circuit (Proposition 19). The completeness parameter is 2−poly(n)and soundness parameter is 1

poly(n).

Finally, we can conclude,

(31)

6 Conclusions and Open Problems

We have shown that the O(logn)-commuting local hamiltonian problem seems to exhibit be-haviours that makes it difficult to comment on its hardness. From section 3, we see that a very symmetric case ofO(logn)-CLHP is indeed in NP, and further from section 4, unlike the non-commuting case, a generalized area law holds for all non-commuting hamiltonians, which seems to suggest a certain limit on entanglement of ground states of commuting hamiltonians, possibly suggesting that thek-CLHP is inQCMA. However, since traversing the hardness ofO(logn)local hamiltonians is as hard ask-local hamiltonians, it might be the case thatO(logn)local commuting hamiltonians (not necessarily geometrically local) are indeedQMA-hard. However, it is difficult to make any specific comments, and we leave the following as open problems:

Open Problem 1:Is traversing the ground space ofk-local commuting hamiltoniansQCMAhard for a constantk?

Open Problem 2: IsO(logn)-CLHPQMAcomplete?

Open Problem 3: Using the generalized area law, is it possible to given an algorithm inQCMA

(32)

References

[1] Dorit Aharonov and Lior Eldar. On the complexity of commuting local hamiltonians, and tight conditions for topological order in such systems. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 334–343. IEEE, 2011. 6, 8

[2] Dorit Aharonov, Daniel Gottesman, Sandy Irani, and Julia Kempe. The power of quantum systems on a line. Communications in Mathematical Physics, 287(1):41–65, 2009. 1, 2

[3] Dorit Aharonov, Aram W Harrow, Zeph Landau, Daniel Nagaj, Mario Szegedy, and Umesh Vazirani. Local tests of global entanglement and a counterexample to the generalized area law. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 246–255. IEEE, 2014. 2, 14

[4] Itai Arad, Alexei Kitaev, Zeph Landau, and Umesh Vazirani. An area law and sub-exponential algorithm for 1d systems. arXiv preprint arXiv:1301.1162, 2013. 2, 11, 12

[5] Itai Arad, Zeph Landau, and Umesh Vazirani. Improved one-dimensional area law for frustration-free systems. Physical review b, 85(19):195145, 2012. 2, 11

[6] Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. 4

[7] Sergey Bravyi and Mikhail Vyalyi. Commutative version of the k-local hamiltonian problem and common eigenspace problem. arXiv preprint quant-ph/0308021, 2003. iv, 1, 2, 3, 6, 7, 8

[8] Stephen A Cook. The complexity of theorem-proving procedures. InProceedings of the third annual ACM symposium on Theory of computing, pages 151–158. ACM, 1971. 1

[9] Sevag Gharibian, Yichen Huang, Zeph Landau, and Seung Woo Shin. Quantum hamiltonian complexity. arXiv preprint arXiv:1401.3916, 2014. 6

[10] Sevag Gharibian and Jamie Sikora. Ground state connectivity of local hamiltonians. In Au-tomata, Languages, and Programming, pages 617–628. Springer, 2015. 3, 15, 18, 19, 23

[11] Parikshit Gopalan, Phokion G Kolaitis, Elitza Maneva, and Christos H Papadimitriou. The connectivity of boolean satisfiability: computational and structural dichotomies. SIAM Jour-nal on Computing, 38(6):2330–2355, 2009. 3

[12] Matthew B Hastings. An area law for one-dimensional quantum systems.Journal of Statistical Mechanics: Theory and Experiment, 2007(08):P08024, 2007. 2

[13] Julia Kempe, Alexei Kitaev, and Oded Regev. The complexity of the local hamiltonian prob-lem. SIAM Journal on Computing, 35(5):1070–1097, 2006. 1, 16, 24

[14] Julia Kempe and Oded Regev. 3-local hamiltonian is qma-complete. arXiv preprint quant-ph/0302079, 2003. 1, 24

[15] Alexei Yu Kitaev, Alexander Shen, and Mikhail N Vyalyi. Classical and quantum computation, volume 47. American Mathematical Society Providence, 2002. 1, 15, 16, 24

(33)

[17] Zeph Landau, Umesh Vazirani, and Thomas Vidick. A polynomial-time algorithm for the ground state of 1d gapped local hamiltonians. arXiv preprint arXiv:1307.5143, 2013. 2

[18] Roberto Oliveira and Barbara M Terhal. The complexity of quantum spin systems on a two-dimensional square lattice. Quantum Information & Computation, 8(10):900–924, 2008. 1

[19] Norbert Schuch. Complexity of commuting hamiltonians on a square lattice of qubits. Quan-tum Information & Computation, 11(11-12):901–912, 2011. 2, 6, 8, 9

[20] Thomas Vidick. Lecture Notes on Hamiltonian Complexity and Quantum PCP Conjecture. http:

//users.cms.caltech.edu/~vidick/teaching/286_qPCP/index.html. 12, 16

[21] John Watrous.Lecture Notes on Quantum Computation.https://cs.uwaterloo.ca/~watrous/ LectureNotes.html. 4

Behavior of O(log n) Local Commuting Hamiltonians

Behavior of

O

(

log

n

)

local commuting hamiltonians

Contents

∑

∑

∑

∑

∑

4

Area Laws

6

Conclusions and Open Problems

References