Quantum Sequential Hamiltonian Algorithm

# Quantum Sequential Hamiltonian Algorithm

Hongye Yu(ä½æ³ç¨) International Center for Quantum Materials, School of Physics, Peking University, 100871, Beijing, China    Yuliang Huang(é»å®äº®) Health Science Center, Peking University, 100191, Beijing, China International Center for Quantum Materials, School of Physics, Peking University, 100871, Beijing, China    Biao Wu(å´é£) International Center for Quantum Materials, School of Physics, Peking University, 100871, Beijing, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
July 5, 2019July 5, 2019
July 5, 2019July 5, 2019
###### Abstract

We propose a generic quantum algorithm, sequential Hamiltonian algorithm, and prove rigorously that our algorithm is as efficient as quantum circuit algorithm. Our quantum algorithm consists of a series of Hamiltonians, , where has a simple and easy-to-construct ground state and the ground state of is the solution. The algorithm works by adiabatically switching on and off the Hamiltonians in sequence. The time complexity of our algorithm is determined by both the number of Hamiltonians and the minimum energy gap during the adiabatic switchings. We give an analytical example where our algorithm has an exponential speed-up over the usual quantum adiabatic algorithm . A heuristic understanding of this speed-up is offered.

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## I Introduction

Quantum algorithms have two paradigms: one is quantum circuit algorithm ChuangBook () and the other is quantum adiabatic algorithm (QAA) Farhi2000 (). The latter works by adiabatically evolving in the ground state of a system with Hamiltonian

 H(s)=(1−s)HB+sHP, (1)

where increases slowly from 0 to 1 as a function of time. The beginning Hamiltonian has a ground state which is easy to construct and the problem Hamiltonian has a ground state that contains the solution of the problem. According to quantum adiabatic theorem, the speed of the algorithm is limited by the minimum energy gap between the ground state and the first excited state during the evolution of . If has an exponentially small minimum gap then the algorithm is inefficient. These two kinds of quantum algorithms are shown to be equivalent polynomially to each other in terms of time complexity Dam1 (); Dam ().

We propose a different algorithm, quantum sequential Hamiltonian algorithm. Our algorithm can be regarded as a natural extension of QAA. However, there are significant differences. First, we can show rigorously that any quantum circuit algorithm can be converted into our quantum sequential Hamiltonian algorithm with the same time complexity. Secondly, we use an analytical example to show that our quantum algorithm can achieve an exponential speed-up over QAA.

Here is how the rest of our paper is organized. We describe quantum sequential Hamiltonian algorithm (QSHA) in Section II. In Section III, we show how a quantum circuit algorithm can be converted into our QSHA with the same time complexity. We then give an example in Section IV where our algorithm has an exponential speed-up over the usual QAA. An intuitive understanding of the exponential speed-up is offered in Section V.

## Ii quantum sequential Hamiltonian algorithm

The quantum sequential Hamiltonian algorithm (QSHA) consists of a series of Hamiltonians

 H0, H1, H2, ⋯, Hj−1, Hj, ⋯, HL−1, HL. (2)

Here and play the same roles of and in the quantum adiabatic algorithm, respectively: has a simple ground state that is easy to create and has the ground state that is the solution of the problem. The algorithm works by preparing the system in the ground state of and then adiabatically turn on and off these Hamiltonians in sequence. Specifically, at the th step (), we switch off and turn on by changing slowly from 0 to 1 in

 Hj(s)=(1−s)Hj−1+sHj. (3)

If the algorithm is successful, after the th step, we end at and obtain its ground state, the solution. For the QSHA algorithm to be efficient, one needs to design the series of Hamiltonians are chosen such that the minimum energy gap between the ground state and the first excited state at any given step is not zero and depends on the system size polynomially , where is the number of qubits. If this is achieved, the time complexity of our algorithm is . We show next that any quantum circuit algorithm can be converted to a QSHA with the same time complexity.

## Iii Proof of exact equivalence to quantum circuit algorithm

Consider a quantum circuit algorithm that has qubits and unitary gates,

 |α0⟩\lx@stackrelU1⟶|α1⟩⋯|αℓ−1⟩\lx@stackrelUℓ⟶|αℓ⟩⋯|αL⟩\lx@stackrelUL⟶|αL⟩, (4)

where represents the th unitary gate operation, . Our aim is to construct a corresponding QSHA with the same time complexity. For this purpose, we introduce additional clock qubits and focus on a special type of clock states which denotes that the first qubits are ones and the rest are zeros Kitaev (); Dam (). Corresponding to the th gate operation, we define an operator

 Oℓ = η2I⊗|ℓ−1⟩c⟨ℓ−1|c−12Uℓ⊗|ℓ⟩c⟨ℓ−1|c (5) −12U†ℓ⊗|ℓ−1⟩c⟨ℓ|c+12ηI⊗|ℓ⟩c⟨ℓ|c,

where . This operator was introduced in Ref.Kitaev (); Dam () for the special case . We construct the following series of Hamiltonians

 H0 = L∑ℓ=1I⊗|ℓ⟩c⟨ℓ|c; (6) H1 = O1+L∑ℓ=2I⊗|ℓ⟩c⟨ℓ|c; ⋯⋯ Hj = j∑ℓ=1Oℓ+L∑ℓ=j+1I⊗|ℓ⟩c⟨ℓ|c; ⋯⋯ HL = L∑ℓ=1Oℓ. (9)

Our algorithm works by turning on and off these Hamiltonians in sequence according to with the initial state .

The ground state energy of all the Hamiltonians ’s is zero. The ground state of is

 |ψηj⟩=√η2−1η2j+2−1j∑ℓ=0ηℓ|γℓ⟩, (10)

where . If our algorithm is successful, we should arrive at , where the probability of finding the solution is , which is independent of the system’s size and can be made very close to one with large . The efficiency of our algorithm depends crucially on the minimum energy gap between the ground state and the first excited state during the entire operation. We next examine this issue.

The whole Hilbert space has a dimension of . However, the subspace of dimension spanned by is invariant during the adiabatic operation. When ’s are used as basis, all the Hamiltonians ’s are essentially tridiagonal matrices of dimension . In fact, has two decoupled parts, and . The first part has the ground state in Eq.(10) with eigen-energy being zero. Its first excited state has an energy larger than according to the Gershgorin circle theorem. The second part has degenerate energy levels with eigen-energy being one. Overall, the minimum gap is the smaller between and one, which are both finite and independent of the system size. Figure 1: The eigenvalues of the ground state and the first excited state for Hj(s) with η=4 and j→∞. The blue lines are calculated from Eqs.(11,12). The dashed line denotes (η+1/η)/2−1=9/8.

It can be shown that for and , the lowest two eigenvalues of satisfy the following equations (see Appendix A for detailed derivation)

 η(1−s)+s22−2s+s/η−2E=exp(α), (11) E=12(1η+η)−coshα. (12)

They have two solutions for , which are plotted in Fig. 1. For close to 0, has two values close to 0 and 1, and for close to 1, has one value close to 0. This is consistent with our above analysis of the eigenvalues of . It is clear from the figure that for every fixed , there is a size-independent gap.

We can now conclude that the minimum gap is independent of the system size during each step of the adiabatic evolution of the Hamiltonians. Therefore, the time complexity of our algorithm is , and we have proven that any given quantum circuit algorithm can be converted to a QSHA with the same time complexity.

It is quite enlightening when our QSHA is compared to the usual QAA

 H(s)=(1−s)H0+sHL. (13)

We can show rigorously that has an exponentially small energy gap at (see Appendix A for details). This implies that our QSHA has an exponential speed-up over the usual QAA. Figure 2: A lattice where each site represents a quantum state |αj⟩. (a) Initial state where a quantum particle residing in a potential well at site 0. (b) The potential well is lifted up slowly at site 0 while another potential well is created adiabatically at site L with increasing depth. (c) The potential well is moved adiabatically site by site while carrying the particle with it.

There is an intuitive physical picture behind the above proof. As shown in Fig.2, the quantum states can be represented by a chain of lattice sites. The initial state is a particle residing in the potential well at site 0. The task is to move the particle to site . One method is to lift up the potential well slowly at site 0 while creating a potential well at site as shown in Fig.2(b). In the end, the well at site 0 disappears completely while a well is created at site . During the process, the particle tunnels through the long chain and arrives at site . This corresponds to the adiabatic algorithm in Eq.(13). Our QSHA corresponds to the method in Fig.2(c), where the potential well carrying the particle is moved adiabatically site by site. This reminds us of the quantum tweezer proposed in Ref. Tweezer ().

The case was studied in Ref.  Dam () to show that any quantum circuit algorithm is polynomially equivalent to a QAA. When , the energy gap is no longer exponentially small and becomes inversely proportional to  Dam (). The constructed QAA is polynomially slower than the corresponding quantum circuit algorithm.

## Iv a special SAT Problem

After the QAA was proposed, it was immediately applied to the 3-SAT problem Farhi2001 () and numerical results indicated that the algorithm could be polynomially efficient. However, a special SAT instance was found by Reichardt Reichardt () as a counter example. The adiabatic Hamiltonian for this SAT instance is given by

 HSAT=−(1−s)n∑k=1σxk−sn−1∑k=1ckσzkσzk+1 (14)

where varies with period that is of order ,

 ck={W>1,If ⌈k/q⌉ is odd1,otherwise (15)

The minimum gap of this Hamiltonian is exponentially small in the period length and the gap goes like Reichardt (); Hermisson (). The adiabatic algorithm is thus not polynomially efficient. We show next that there is a QSHA for this SAT instance with time complexity O(n).

For this SAT instance, we construct the QSHA with the following Hamiltonians

 H0 = −n∑k=1σxk, (16) Hj = −n∑k=j+1σxk−j−1∑k=1ckσzkσzk+1, (17) Hf = −n−1∑k=1ckσzkσzk+1 (18)

where . It can be shown that every has a finite gap that is independent of the system size . Figure 3: The solid lines are the eigenvalues of hj(s) with cj=3 and the dashed lines are the eigenvalues of H3(s) for 5 bits with cj=j. The smallest two eigenvalues of the H3(s) can be found simply by adding (∑2k=1±ck+∑5k=5±1) from the eigenvalues of hj(s). It is clear that if hj(s) has a size-independent gap, then the corresponding Hj(s) also has a size-independent spectral gap.

We write out all terms in

 Hj(s) = −N∑k=j+2σxk−j−1∑k=1ckσzkσzk+1 (19) −(1−s)σxj+1−scjσzjσzj+1.

We immediately notice: (1) there are only two -dependent terms and they contain just spin and spin ; (2) the other two terms do not share any spin with each other and with the two -dependent terms. Therefore, the eigenstates of this Hamiltonian must be of the following form

 |±1z⟩1...|±1z⟩j−1|X⟩j,j+1|±1x⟩j+2...|±1x⟩N, (20)

where and are the eigenstates of and with the eigenvalues , respectively. is an eigenvector of

 hj(s)=(1−s)σxj+1−sσzjσzj+1. (21)

In the basis of and , we have

 hj(s)=⎛⎜ ⎜ ⎜ ⎜⎝−cjs0s−100cjs0s−1s−10cjs00s−10−cjs⎞⎟ ⎟ ⎟ ⎟⎠. (22)

It has two eigenvalues , which have a gap independent of the system size as shown in Fig. 3. We can get all the eigenvalues of by adding , which are also shown in Fig. 3. Therefore, we have a size-independent gap in every and our QSHA is of time complexity , which is an exponential speed-up over .

## V Discussion and Conclusion

Let us review the QAA in Eq.(1). The ground state of usually is an equal-weight superposition of all possible states. For simplicity we assume that there is only one solution . So, we have . This is exactly the reason that the QAA for unsorted search has a minimum gap of and, therefore, is of time complexity  Dam1 (); Cerf2000 (); Hu (). In this perspective, the exponentially small gap for Eq.(1) is not accidental but rather a rule.

Usually an exponentially small energy gap signals a “quantum phase transition”: there is a dramatic change to the ground state wave function around , the point where the exponentially small gap occurs. The change is marginal before and after . In other words, the algorithm embodied in Eq.(1) puts its entire workload in a very small interval around . If we could spread out the workload evenly in the entire process, we would be able to speed up the algorithm. Our QSHA does precisely this kind of spreading. Suppose is the ground state of . If we find an efficient QSHA where , with being independent of , we have

 (23)

This shows that an exponentially small gap is split into the product of polynomially number of small gap . Interestingly, as each step from to takes about the same time , the amount of time for the entire process is about , which is an exponential speed up over , the running time for the QAA.

Note that quantum phase transition is a well known subject in many-body physics Sachdev (). In the above, by “quantum phase transition” we meant a dramatic change in the ground state wave function at a critical point. It can be for either a single-particle system or a many-particle system.

The advantage of our QSHA over QAA can be put in another perspective. In our QSHA, there is much bigger freedom to design an algorithm. In QAA, the algorithm is fixed once and are chosen. In contrast, in QSHA, with fixed and , we have enormous amount of different sequences to choose from. We expect that more faster quantum algorithms be found in the form of QSHA.

In sum, we have presented quantum sequential Hamiltonian algorithm (QSHA). We have proved rigorously that any quantum circuit algorithm can be converted to a QSHA with the same time complexity. Furthermore, we have used two analytical examples to show that our QSHA can have an exponential speed-up over the usual quantum adiabatic algorithm. Our work shows that designing an efficient quantum algorithm can now be entirely an endeavor of physics.

## Vi acknowledgement

This work was supported by the National Basic Research Program of China (Grants No. 2013CB921903) and the National Natural Science Foundation of China (Grants Nos. 11334001 and 11429402).

## Appendix A Analytical results of energy gaps

In this Appendix we give detailed derivations of two mathematical results regarding minimum energy gaps used in Section III. We present these results in a self-contained form so that they can be read without knowing anything in our main text.

We define three () matrices, , , and . The matrix is diagonal with and (). The matrix is tridiagonal with

 M11=η/2, Mmm=η+1/η2,     (2≤m≤N−2) M(N−1)(N−1)=12η, (24) MNN=1, Mm(m+1)=M(m+1)m=−12,   (1≤m≤N−2) M(N−1)N=MN(N−1)=0.

The matrix is also tridiagonal with

 F11=η/2, Fmm=η+1/η2,     (2≤m≤N−1) FNN=12η, (25) Fm(m+1)=F(m+1)m=−12,   (1≤m≤N−1).

In the following discussion, we assume that .

### a.1 Finite gap

We consider the matrix with . We are interested in its lowest two eigenvalues. Assume that has an eigenvalue with a corresponding eigenvector . The eigen-equation

 H1(s)X=λX (26)

can be explicitly written as

 η2x1−12x2 = λx1, −12xk−1+12(1η+η)xk−12xk+1 = λxk, −12xN−2+12(1η+ηs)xN−1−12sxN = λxN−1, −12sxN−1+(1−s+12ηs)xN = λxN.

where . We can eliminate the and have

 η2x1−12x2 = λx1, −12xk−1+12(1η+η)xk−12xk+1 = λxk, −12xN−2+12[1η+2ηs(1−s−λ)2−2s+s/η−2λ]xN−1 = λxN−1.

We introduce two additional variables and . These equations are equivalent to the standard second order difference equation

 xk−1−(1η+η−2λ)xk+xk+1 = 0, (29) xN−2−(1η+η−2λ)xN−1+~xN = 0, (30)

where with the boundary conditions

 ηx0=x1, (31) [η(1−s)+s22−2s+s/η−2λ]xN−1=~xN. (32)

It has two types of solutions. Type I is given by

 xk=Asin(kα)+Bcos(kα), (33)

with . Type II is given by

 xk=Asinh(kα)+Bcosh(kα), (34)

with . The value of and, therefore, the eigenvalue are determined by the two boundary conditions.

Note that if is a solution with the eigenvalue , then is the same solution with the same . In addition, if , then we will find and all the vanish. So without loss of generality we only consider the situation.

The matrix are two eigenvalues of zero and one with corresponding eigenvectors being and , respectively. For , the type I eigenvalue is . The smallest two eigenvalues of must be of type II, satisfying the equation (34). For , we have

 sinh(Nα)∼cosh(Nα)∼12exp(Nα). (35)

Therefore, in the limit of , the two lowest eigenvalues are determined by the following equations

 η(1−s)+s22−2s+s/η−2λ=exp(α), λ=12(1η+η)−coshα. (36)

As the size of the matrix does not enter the above two equations, it is clear that the gap is bounded from the below by a constant. Our calculation with the above equations shows that the minimum gap is around at .

### a.2 Exponentially small energy gap

We consider another Hamiltonian with . We shall show that the gap between the lowest two eigenvalues of this Hamiltonian is exponentially small as at

 s∗=2η−1/η+2. (37)

At , can be written as

 H2(s∗)=s∗⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝η2−120⋯⋯0−12η−120⋯⋮0−12η−12⋱⋮⋮⋱⋱⋱⋱⋮⋮⋯⋱−12η−120⋯⋯0−12η2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (38)

Since is a constant independent of , we can just discuss the gap of . Assume that has an eigenvalue and an eigenvector that satisfy

 H∗2X=λX. (39)

We write the above equation in its component form as

 η2x1−12x2 = λx1 −12xk−1+ηxk−12xk+1 = λxk, (40) −12xN−1+η2xN = λxN

where . By introducing two additional variable and , we can convert the above equations into the standard second order difference equation

 xk−1−2(η−λ)xk+xk+1=0, (41)

where and with the boundary conditions

 x0=ηx1,  ηxL=xL+1 (42)

It has two types of solutions. Type I solution is given by

 xk=Asin(kα)+Bcos(kα), (43)

with . Type II solution is given by

 xk=Asinh(kα)+Bcosh(kα), (44)

with . The two boundary conditions determine the value of and .

With a similar argument as in the last subsection, we only consider the situation. For , type I eigenvalue . However, according to the Gershgorin circle theorem, has and only has two eigenvalues smaller than . Therefore, the smallest two eigenvalues are of type II. For the type II solution, the boundary conditions are

 B=η(Asinhα+Bcoshα), (45)

and

 η(AsinhNα+BcoshNα) (46) = Asinh(N+1)α+Bcosh(N+1)α.

After eliminating and we have

 ηsinhα1−ηcoshα=sinh(N+1)α−ηsinhαηcoshNα−cosh(N+1)α, (47)

which can be simplified into

 η2sinh(N−1)α−2ηsinhNα+sinh(N+1)α=0. (48)

Let , we can rewrite the equation as follows

 z2−2ηz+η2−z−2(N−1)(η−1/z)2=0. (49)

As , we have . For convenience, we define

 f(z)=z2−2ηz+η2−z−2(N−1)(η−1/z)2. (50)

Also note that in the following discussion we alway have and .

It is easy to find , , , and . Moreover, has at most 2 roots, for the has and only has 2 eigenvalues satisfying the equation (44). Thus there is one root in the interval and in . So if , we can find a satisfying . Let , and we can find the roots are within the internal . The gap can be bounded by the distance between and

 |λ1−λ2|=12|z1−z2+1/z1−1/z2|<12|z1−z2|. (51)

where is no more than . Furthermore, by increasing the system’s size , we can let be arbitrarily small. So we can find a satisfying for all and arbitrary . Therefore for , are within the internal . Note that for , , which is exponential with . Thus we come into the conclusion: For , there exist a and for all , the gap of is smaller than , where is an arbitrarily small constant.

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