Improvement of quantum walks search algorithm in single-marked vertex graph

Quantum walks are powerful tools for building quantum search algorithms. However, these success probabilities are far below 1. Amplitude amplification is usually used to amplify success probability, but the soufflé problem follows. Only stop at the right step can we achieve a maximum success probability. Otherwise, as the number of steps increases, the success probability may decrease, which will cause troubles in the practical application of the algorithm when the optimal number of steps is unknown. In this work, we define generalized interpolated quantum walks instead of amplitude amplification, which can not only improve the success probability but also avoid the soufflé problem. We choose a special case of generalized interpolated quantum walks and construct a series of new search algorithms based on phase estimation and quantum fast-forwarding, respectively. Especially, by combining our interpolated quantum walks with quantum fast-forwarding, we both reduce the time complexity of the search algorithm from Θ((ε−1)HT) to Θ(log(ε−1)HT) and reduce the number of ancilla qubits required from Θ(log(ε−1)+logHT) to Θ(loglog(ε−1)+logHT) , where ɛ denotes the precision and HT denotes the classical hitting time. In addition, we show that our generalized interpolated quantum walks can improve the construction of quantum stationary states corresponding to reversible Markov chains. Finally, we give an application to construct a slowly evolving Markov chain sequence by applying generalized interpolated quantum walks, which is the necessary premise in adiabatic stationary state preparation.


Introduction
Algorithms based on quantum walks have been studied widely and can speed up classical random walk algorithms on many problems, such as element distinctness [1], triangle finding [2], matrix product verification [3], especially on searching problems [4] and sampling problems [5].
In classical random walks, hitting time describes the speed to reach the marked vertex set and characterizes the algorithm complexity of the classical search problem [6,7]. Since quantum walks have shown their advantages compared with random walks, the studies in the quantum version of the search algorithm are the current research focus. By quantizing classical discrete Markov chains, a bipartite quantum walk model was proposed by Szegedy [8], which proved that quantum algorithms quadratically speed up classical hitting time in detecting the existence of marked vertices for ergodic and reversible Markov chains. After Szegedy's framework, a series of new algorithms with better complexity perform well in many variety contexts [5,9,10].
By introducing recursive amplitude amplification, Magniez et al extended Szegedy's framework to find marked vertex in reversible ergodic Markov chains [11]. However, it can not quadratically speed up the classical search problem in any graph, such as the 2D grid. Tulsi proposed a new technique to solve the open problem [12]. By extending Tulsi's technique, Magniez et al [13] demonstrated the possibility of finding a unique marked vertex with a square root speed up over the classical hitting time for any reversible state-transitive single-marked vertex Markov chain. In [14], Krovi et al introduced the interpolated quantum walks, the serious restrictions are relaxed to a reversible Markov chain with a single-marked vertex. Recently, Ambainis [15] introduced a search algorithm for any reversible Markov chain with quadratic speedup, which is more suitable for multi-marked vertices graphs but does not perform as well as [14] on single-marked vertex graphs.
However, all the above search algorithms can only achieve constant success probability or even less, and to improve success probability, amplitude amplification is usually introduced. Unfortunately, the implementation of amplitude amplification often introduces the soufflé problem [16] because the largest success probability can be achieved only when we stop at the right time. Stopping too early or too late will not maximize the success probability. Since the success probability has only a lower bound instead of an exact value, which means we do not know the proper time to stop. When we stop late, the success probability will decrease as the number of steps increases. Besides, now all search algorithms with constant success probability are linearly related to ε −1 , where ε denotes the error of the algorithm. The dependency on error is too large.
To increase success probability, in this work, we introduce generalized interpolated quantum walks, which will be combined with phase estimation to construct a new search algorithm instead of amplitude amplification. The introduction of generalized interpolated quantum walks both amplifies the success probability and avoids the soufflé problem while maintaining the time complexity.
Based on reversible Markov chains, quantum fast-forwarding algorithm [17] can simulate actions of n random walk steps by √ n quantum walk steps. Here we find generalized interpolated quantum walks can be combined with quantum fast-forwarding. The combination reduces the dependency on error, which not only reduces the time complexity but also reduces the number of ancilla qubits required.
Except for the search problem, another fundamental problem in the classical random walk is the sampling from the stationary distribution of the Markov chain. Google's PageRank algorithm [18] can be reduced to prepare the stationary distribution. Currently, the qsampling (quantum sampling) algorithms which achieve the quadratic speedup compared with the classical case are only known for some limited stationary distributions [19,20] and a series of sparse graphs [14,21]. In [22] we provide a new discrete-time quantum walk-based sampling algorithm in cost Θ( √ HT log 1 ε ), which is suitable for any reversible Markov chain with no additional requirement on graphs or stationary distributions and achieves quadratic speedup in a series common graphs such as sparse graphs. However, similar to the above search algorithm, qsampling algorithms face the soufflé problem. Our generalized interpolated quantum walks can improve the success probability and avoid the soufflé problem as well.
Finally, we give an application of generalized interpolated quantum walks. In adiabatic quantum computing, a quantum state can be prepared by preparing the ground state of a slowly evolving Hamiltonian sequence. The ground state of Hamiltonian can be seen as the stationary state of the corresponding Markov chain. However, it is not easy to prepare the needed Hamiltonian or Markov chain in adiabatic quantum computing. By applying generalized interpolated walks, we can construct a series of slowly evolving Markov chains and prepare a quantum stationary state in adiabatic quantum computation.
The paper is organized as follows. Preliminaries are provided in section 2.1 firstly. Second, we define generalized interpolated walks in section 2.2. Then to amplify the success probability of search algorithms and qsampling algorithms, we combine generalized interpolated walks with quantum phase estimation and quantum fast-forwarding instead of amplitude amplification in sections 3.1 and 3.2. In addition, we apply generalized interpolated walks to prepare the quantum stationary state of Markov chains in adiabatic quantum computing in section 3.3. Finally, the paper is concluded in section 4.

Preliminaries
Consider an undirected graph G(V, E), where V is the set of vertices and E is the set of edges. G consists of N = |V| vertices. A random walk on G can be seen as a Markov chain P with the corresponding transition matrix P = (p xy ) x,y∈V , where p xy denotes the transition probability from x to y for arbitrary vertex x, y ∈ V [23]. In the following, we will use the terms 'random walk', and 'Markov chain' interchangeably.
A Markov chain is irreducible if every vertex is reachable from any other vertex. A Markov chain is aperiodic if the length of each directed cycle has the greatest common factor of 1. If a Markov chain is irreducible and aperiodic, then we call it ergodic. By Perron-Frobenius theorem [24], each ergodic Markov chain has a unique 1-eigenvector (the eigenvector corresponding to eigenvalue 1), i.e. stationary distribution, which is represented as π = (π x ) x∈V . An ergodic Markov chain is reversible if it satisfies the detailed balance equation π x p xy = π y p yx , ∀x, y ∈ V. (1) All eigenvalues of reversible Markov chains lie in [−1, 1] [23]. Since lazy walks can be introduced and set as P lazy = (P + I)/2 [14], here we assume that all eigenvalues of reversible Markov chains lie in [0, 1] without loss of generality. Let λ 0 denote the eigenvalue of P that equals to 1 and the other eigenvalues λ j for j = 1, . . . , n − 1 are strictly less than 1 in nonincreasing order. The eigenvalue gap of P is ∆ := 1 − λ 1 .
By quantizing classical Markov chains, Szegedy [8] firstly developed the theory of quantum walk search algorithms based on the bipartite graph constructed by the original graph. The bipartite graph corresponds to two quantum registers and the walker walks alternately between the two registers. In detail, quantum walks perform on the extended Hilbert space which is a superposition state that simulates the edges outgoing from the vertex x in one side to the other side of bipartite graph according to the original connecting relation. Similarly, define which is a superposition state that simulates the edges outgoing from the vertex y in the other side of bipartite graph. Szegedy's quantum walk operator consists of two reflections as follows: Discriminant matrix is often considered when studying spectral properties of Markov chain P, which is defined as For any reversible Markov chain P, by balance equation (1) we have D(P) = diag(π) 1/2 · P · diag(π) −1/2 , where diag(π) denotes the diagonal matrix with (diag(π)) xx = π x , ∀x ∈ V. Naturally, D(P) has the same spectral gap with P and D(P)|π = |π , where |π := x √ π x |x is the quantum stationary state. Let cos θ 1 , cos θ 2 , . . . , cos θ l be eigenvalues of D(P), the eigenvalues of Szegedy's walk operator can be expressed as e −2iθ1 , e 2iθ1 , . . . , e −2iθ l , e 2iθ l [8].
The Szegedy's walk operator can be rewritten as: where |0 is some fixed state in Λ V , unitary operator V(P) satisfies that Krovi et al [14] defined the quantum walk operator W(P) as In Szegedy's walk, the one-step walk corresponds to (V(P) · W(P) · V(P † )) 2 , and t-steps walk corresponds to V(P) · W(P) 2t · V(P † ), namely Szegedy's walk operator and Krovi's walk operator have the similar properties.
2.1.1. Hitting time. Assume a vertex subset M ⊆ V is marked in graph G(V, E), and then the search problem is to find any vertex in M by walking in the graph G. Hitting time is defined as the smallest number of steps required to hit any marked vertex in M for a random walk P that starts from the stationary distribution π, which is denoted by HT(P, M) and satisfies the following property [14]: √ π x |x , λ ′ k and |v ′ k are eigenvalues and the corresponding eigenvectors of the discriminant matrix D(P ′ ), and P ′ is the absorbing version of the Markov chain P that all outgoing transitions from marked vertices are replaced by selfloops. Let HT := max x∈V HT(P, {x}) denote the maximum hitting time to reach any vertex in Krovi V et al introduced interpolation into the search algorithm based on the Markov chain firstly [14,25]. The interpolated Markov chain P(s) can be expressed as where 0 ⩽ s ⩽ 1, and the corresponding W(s) and D(s) are defined as W(P(s)) and D(P(s)), respectively. For s ∈ [0, 1), P(s) is an ergodic and reversible Markov chain. Then P(s) has the unique 1-eigenvector |v 0 (s) with eigenvalue λ 0 (s) = 1, and the other eigenvalues λ j (s) for j = 1, . . . , n − 1 are strictly less than 1 in non-increasing order with corresponding eigenvector |v j (s) . The interpolated hitting time HT(s) is defined as for any s ∈ [0, 1), where λ k (s) and |v k (s) are eigenvalues and the corresponding eigenvectors of the discriminant matrix D(P(s)). Quantum walk algorithms considered here are based on eigenvalue estimation performed on operator W(S). There may be more than one 1eigenvectors for W(s), and we choose one of them as |v 0 (s) to introduce the following results.
Define |v 0 (s) as where Then for ∀s ∈ [0, 1], we have The detailed relationship between W(s) and D(s) is listed as follows:

Lemma 1 (spectrum of W(s) and D(s) [8]). The eigenvalues and eigenvectors |v k (s) of D(s) satisfy that
for k = 0, . . . , n − 1, with φ 0 (s) = 0, the eigenvalues and eigenvectors of W(s) are: , and the relation between them can be expressed as:

Search problems.
Quantum walk-based search algorithms are usually constructed by the following process. Assume that quantum walks have access to the following (controlled) unitary operators, where the detailed construction can be seen as an oracle:

Update(P): implements a (controlled) walk operator W(P). Complexity U.
Check(M): checks whether x is a marked vertex. Complexity C. Described by the mapping In general, a quantum walk search scheme is mainly based on phase estimation [11,14,26] or quantum fast-forwarding [17]. The search algorithm [14] evolved by interpolated walks and phase estimation can be expressed as follows: a. If it is marked, measure and output the current state. b. Otherwise, apply is quantum phase estimation operator and expressed in the following lemma.

Lemma 2 (phase estimation [26]). Let A be an arbitrary unitary operator on n qubits and |Ψ k be any eigenvector of
It does not need to know the exact value of A or |Ψ k or e iϕ k . For any precision t ∈ N, there exists a phase estimation

The algorithm calls O(2 t ) controlled-A operator and O(t) ancilla qubits in total.
For any reversible Markov chain with single-marked vertex g, after the Check(M) in step 2, if the marked vertex is measured, we get |g and the algorithm stops; else we have |π and apply quantum phase estimation operator U E (W(s), log √ HT ). The phase estimation operator U E distinguishes 1-eigenvector from other eigenvectors by marking the ancilla registers. By choosing an appropriate interpolated parameter s, |v 0 (s) has constant overlap with both |π and the marked state |g , while U E keeping the part of |v 0 (s) and reducing the other part. Then the output state in the above search algorithm has constant overlap with the target state |g , but the maximum success probability is about 1/4 − ε. Amplitude amplification is usually applied to enlarge the success probability. The output state in step 3 can be expressed as Amplitude amplification is composed of reflection around |ψ and reflection around the target state |g . The former can be constructed by two unitary operators and one reflection as follows: where Rπ = I − 2|π π|, and the latter can be constructed by Check(M). However, amplitude amplification will bring the soufflé problem, where the largest success probability is achieved only when stopping at the right time. The success probability has only a lower bound instead of an exact value and we do not know the proper time to stop.
The above search algorithm is based on the quantum phase estimation. In the following, we introduce quantum fast-forwarding, the other method recently used to construct the search algorithm. The quantum fast-forwarding method is introduced by Apers and Sarlette [17] to simulate the dynamics of classical random walk. For quantum walk operator W and positive integer τ , the detailed construction of quantum fast-forwarding operator U F (W, τ ) is described in the following lemma: Lemma 3 (quantum fast-forwarding [17]). For any reversible Markov chain and corresponding walk operator W(P) on state space Λ V ⊗ Λ V and any |ψ ∈ Λ V , quantum fast-forwarding algorithm U F (W, τ ) acting on |ψ |00 τ will output a state ε 0 -close to D t |ψ |00 τ + |χ |00 ⊥ , where |00 ⊥ satisfies 00 τ |00 ⊥ = 0. The algorithm invokes controlled-W operator Γ := Θ( t log(ε −1 0 )) times and requires τ = log Γ ancilla qubits.
The dynamics of the discriminant matrix D are simulated by the unitary operator U F (W, τ ), which acts as follows: where the detailed description of p l can be found in equation (6) of [17].
Ambainis's quantum algorithm [15] introduced the quantum fast-forwarding method to solve the search problem with the success probability smaller than constant. Then amplitude amplification is required and the soufflé problem will appear [16].
In order to overcome the soufflé problem, we propose a concept of generalized quantum interpolated walks, which can replace amplitude amplification and improve the success probability of the search algorithm to at least 4/5.

Generalized quantum interpolated walks
Here we define generalized interpolated walks as follows.
Definition 1 (generalized quantum interpolated walks). Let P be a Markov chain with a set of marked vertices M, and the corresponding absorbing walk is P ′ . For a series of parameter {s 1 , s 2 , . . . , s t : 0 ⩽ s 1 , . . . , s t ⩽ 1} =: S, an independent Markov chain Q is defined to express the transition probability q ij from s i to s j . s k1 , s k2 , . . . is a sequence determined by the Markov chain Q, where s ki+1 is the next item of s ki according to (q ij ) 1⩽i,j ⩽t . Generalized interpolated Markov chain is defined as P(S) All the previous quantum interpolated walks can be seen as special cases of generalized interpolated walks. We summarize and list them in table 1 and apply a special case in the following sections.

Remark 1 (the relationship between interpolated walk framework).
We list all above interpolated walks as follows, and the relationship between them refers to figure 1. If t = 1, generalized quantum interpolated walks degenerate to Krovi's interpolated walks [14], which are determined by only one parameter s chosen by preparing a 1-eigenvector of W(s) that has a large overlap with both the initial state and the target state.  [14] t = 1 Ambainis's interpolated walks [15] q i,j = 1/t, ∀1 ⩽ i, j ⩽ t Our improved interpolated walks for searching and qsampling Ambainis's interpolated walks [15] can be seen as that each s i in S := {s 1 , s 2 , . . . , s t : 0 ⩽ s 1 , . . . , s t ⩽ 1} is chosen with equal probability and W(s i ) acts on the initial state. Therefore the walk based on Markov chain Q is in a complete graph, and the corresponding distribution keeps Our improved interpolated walks are applied in the following fix the order of each s i in S := {s 1 , s 2 , . . . , s t : 0 ⩽ s 1 , . . . , s t ⩽ 1}. By implementing the walk operator corresponding to each s i one by one, as i increases, the overlap between the 1-eigenvector of the current quantum walk operator and the target state will become larger.
As far as we know, quantum interpolated walks that applied previously to search problems or qsampling problems have only one parameter s, and the maximum success probability corresponding to single implementation is a small constant [14] or even smaller [15].

Algorithms based on quantum phase estimation
By applying generalized interpolated walks instead of amplitude amplification, we will amplify the success probability of the search algorithms and qsampling algorithms while maintaining the time complexity and avoiding the soufflé problem. Since the success probability of existing algorithms is still far from 1, generally, amplitude amplification is used to enlarge the success probability, and then we often face the soufflé problem. When the number of search steps is greater than the steps required, the success probability of these algorithms may drop dramatically as figure 2. Here we replace the single parameter s with S := {s 1 , s 2 , . . . , s t }, the set of parameters, and use this to amplify the success probability instead of amplitude amplification.

Theorem 1 (search marked vertex based on quantum phase estimation).
For an ergodic reversible Markov chain P in G(V, E) with single marked vertex g, algorithm 1 achieves a state ε-close to |g with success probability more than 4/5 from initial state |π . The complexity is S + Θ(ε −1 √ HT)(U + C) with Θ(log(ε −1 ) + log √ HT) ancilla qubits. Intuitively, our quantum algorithm works as figure 3. We fix the set of parameters as {s 1 , s 2 , . . . , s t : 0 ⩽ s 1 ⩽ · · · ⩽ s t ⩽ 1}. Firstly, we map |π to |v 0 (s 1 ) by quantum walk operator based on P(s 1 ), namely keep the part of |π that matches 1-eigenvector |v 0 (s 1 ) and reduce the amplitude of other eigenvectors to ε/t. Similarly, we map |v 0 (s 1 ) to |v 0 (s 2 ) by quantum walk operator based on P(s 2 ), . . . , finally, we map |v 0 (s t−1 ) to |v 0 (s t ) by quantum walk operator based on P(s t ) and measure |v 0 (s t ) in the standard basis to get the marked vertex. Now we analyze the success probability first. For the initial state |π and target state |g , let p succ denote the success probability of algorithm 1, then we have When M = {g}, the state after step 2 can be expressed as |π : from (4), we have Let |v 0 (s 0 ) := |π . For i = 1, . . . , t, and k = 0, . . . , n − 1, define Since for any s ∈ [0, 1) we have |π ∈ Λ V = span{|v k (s) |k = 0, . . . , n − 1}, then For i = 1, . . . , t, define A i as Here we give an estimation of the success probability p succ as below.
which is achieved when Now we bound the value of A i . Define since |v k (s i ) and |v 0 (s i ) are eigenvectors of D(s i ) and corresponds the different eigenvalues, we have from the definition of α i k,0 and α i k,t+1 in (14) = cos Next, we determine the value of A i . From lemmas 1 and 2, since W(s i ) is a real operator, U E (W(s i ), τ ) acts on the eigenvectors of W(s i ) as: Thus, eigenvectors of D(s i ) after phase estimation are Now we bound A i as by the norm invariance after unitary operator, and the norm non-increasing after projection (3), the interpolated hitting time in the single-marked vertex graph is By combining (16) and (20) we have From theorem 17 in [14], we have HT(s i ) ⩽ HT in the single-marked vertex Markov chain.
Since t ⩾ 1, x ⩾ 4, then y ∈ (0, 1], h ′ (y) > 0 and h(y) is an increasing function, which means for any t ⩾ 1, From the definition of f (t), the success probability increases and tends to 1 as the number of steps t increases, which means our method will never face the soufflé problem. When t ⩾ 10 and ε is small, we have p succ ⩾ 4/5, and the success probability increases as t increases. Since , the time complexity of algorithm 1 is S + Θ(ε −1 √ HT)(U + C), and the number of ancilla qubits is τ = Θ(log Γ 1 ) = Θ(log(ε −1 ) + log √ HT).

Algorithms based on quantum fast-forwarding
In the previous section, a new search algorithm based on generalized interpolated walks and phase estimation improves the success probability to nearly 1. However, it relies on ε −1 too heavily that when the value of ε is small, the algorithm complexity also increases rapidly. Quantum fast-forwarding as the quantum version of the random walk with acceleration can be used to reduce the dependence on ε. In this section, we combine the generalized interpolated walks with quantum fast-forwarding, which both reduces the times of calling walk operators and the number of ancilla qubits. Quantum fast-forwarding algorithm is introduced in lemma 3, which describes the dynamics of the discriminant matrix D by the unitary operator U F (W, τ ) for some positive integer τ .
Then our new algorithm for searching is described as follows.
Proof. The search algorithm that is based on the generalized interpolated quantum walks and quantum fast-forwarding is listed as follows.
The unitary operator U S (W(s), τ ) we first defined in [22] is to prepare the quantum stationary distribution. Let U F (s) denote the quantum fast-forwarding operator U F (W(s), τ ) ⊗ I, we have: The symbols above are defined as below: for any operator A, A x represents A acting on the xth register; cA x,y represents controlled-A gate, if the xth register is |1 , A acting on the yth register; similarly, ccA x,y represents controlled-A gate, if the xth register is |0 , A acts on the yth register. The corresponding circuit refers to figure 4. We analyze the success probability first. For initial state |π and target state |g , let p succ denote the success probability, then we have by the norm non-increasing after projection, by triangle inequality and (24) by triangle inequality and (24) Since the ith qubit in the fifth register changes from |0 to |1 only if the state before ccX 234,γi is expressed as | · · · |00 τ 0 |1 i−1 0 t−i+1 , we have since they are applied on the different qubits by the norm invariance after unitary operator since they are applied on the different qubits by the norm invariance after unitary operator the γ i th qubit turns from |0⟩ to |1⟩ only when the quantum state before ccX 234,γi is | · · · ⟩|00 τ 0⟩|ω i−1 ⟩ by the norm non-increasing after projection, From triangle equality, (25) can be expressed as where T satisfies T log(ε −1 0 ) = Θ(Γ 2 ) from lemma 3. From lemma 1 and (5) we have where ∆(s i ) denotes the eigenvalue gap of P(s i ), namely ∆(s i ) : then (26) can be expressed as by |v k (s i )⟩, . . . , |v k (s i )⟩ are mutually orthogonal and the inequality of arithmetic and geometric means From (13) we have namely p succ ⩾ cos 2(t+1) ( π 2(t+1) ) − ε. From the definition of f (x) in (21), success probability increases as the number of steps t increases, so the algorithm avoids the soufflé problem that amplitude amplification will bring. Now we give an algorithm complexity analysis. The time complexity of algorithm 2 is S + Θ(Γ 2 )(U + C). From lemma 2 in [22], Θ(1/∆(s)) ⩽ Θ(HT) for s ∈ [0, 1). Then √ HT), and the time complexity The number of ancilla qubits is τ + 2 = Θ(log Γ 2 ) = Θ(log log(ε −1 ) + log √ HT). Compared with the result in theorem 1, we achieve a faster search algorithm with fewer ancilla qubits.
In the algorithm based on quantum walks, in addition to the search problem, the sampling problem of stationary distribution is often encountered in the real world. For any reversible Markov chain, quantum sampling for stationary distribution is to prepare the quantum state with amplitudes arbitrarily close to the square root of the stationary distribution.
The quadratic speedup for quantum sampling algorithms is only known for some special graphs [21] or special stationary distributions [19]. No universal results are obtained for the more general graphs corresponding to any reversible Markov chain. In [22], we provide a new discrete-time quantum walk-based sampling algorithm in cost Θ( √ HT log 1 ε ), which is suitable for any reversible Markov chain with no additional requirement on graphs or stationary distributions and achieves quadratic speedup on a series of common graphs and sparse graphs. However, similar to the search problem, the success probability of the sampling problem is not large enough and still depends on amplitude amplification, and then it will meet the soufflé problem. Here we list the quantum sampling algorithm [22] and the corresponding amplitude amplification first, and then provide the improved sampling results through generalized interpolated quantum walks.
The quantum sampling algorithm [22] consists of quantum interpolated walks and quantum fast-forwarding algorithm and can be expressed as follows: 1. Choose a vertex g ∈ G randomly and use Setup(g) to prepare |g . 2. Apply U S (s, HT), which can be expressed as ccX 2 which has constant overlap with the target state |π . However, similar to the above search scheme, the success probability is not high enough and amplitude amplification is usually applied to enlarge the success probability. Amplitude amplification is composed of reflection around |ψ and reflection around |π . The former can be constructed as follows: HT), and the latter is constructed in [22], then the soufflé problem will exist.
We introduce generalized interpolated walks to the quantum sampling problem. For a reversible Markov chain, if we set the marked vertex as the initial vertex and reverse the search algorithm, we can obtain a qsamping algorithm. Reversing the above algorithm results, we get a qsampling algorithm as shown below.
Corollary 1 (qsampling based on generalized interpolated walks). For an ergodic reversible Markov chain P in G(V, E), there exists an algorithm that outputs a state ε-close to |π with success probability more than 4 5 from the initial state |g for any g ∈ V. The complexity is S(g) + Θ(log(ε −1 ) √ HT)(U + C), with Θ(log log(ε −1 ) + log √ HT) ancilla qubits.

Application
In this section, we give an application of generalized interpolated walks. Adiabatic quantum computing is an important quantum computing model that is equivalent to standard quantum circuit model [27]. We apply generalized interpolated walks to prepare quantum stationary state of Markov chains for adiabatic quantum computing. In adiabatic quantum computing, the state preparation is the premise of algorithm and is always expressed as a ground state of the target Hamiltonian which can be obtained by a series of slowly changing Hamiltonian. However, the slowly evolving Hamiltonian sequence is not always easy to prepare. For those Hamiltonian that corresponds to the reversible Markov chains, the corresponding ground state can be equivalent to the stationary state [27], 5 and the construction of slowly evolving Hamiltonian sequences are reduced to construct a series of slowly evolving Markov chains with their stationary states [5], where the algorithm to prepare Boltzmann-Gibbs distribution has been constructed. However, the construction of a general slowly evolving Markov chain sequence is not given in [5], although it is necessary when we want to prepare the stationary state of any reversible Markov chain.
By applying generalized interpolated walks, we give the detailed construction of slowly evolving Markov chains to approach any reversible Markov chain P as follows.
The walk operator can be defined as According to the construction of |π(s i ) we know: For any i ∈ {0, 1, . . . , t}, let and we have |v 0 (s i ) = cos θ i |π + sin θ i |0 = cos( iπ 2(t+1) )|π + sin( iπ 2(t+1) )|0 . Now we prove that all the above Markov chains satisfy varying slowing conditions, which means that for any two adjacent Markov chains, there exists q such that the inner product of their quantum stationary state is greater than q.
If t ⩾ π 2 arccos q − 1, then cos( π 2(t+1) ) ⩾ q, and the slowing evolving condition π i |π i+1 ⩾ q is satisfied for i ∈ {0, 1, . . . , t}, where the quantum stationary state of the first Markov chain is easy to prepare and the quantum stationary state of the last Markov chain approximates target state.

Discussion
In this work, we define generalized interpolated walks and apply them instead of amplitude amplification, which both improves the success probability of the search algorithm and the qsampling algorithm and avoids the soufflé problem. By introducing quantum fast-forwarding, we not only reduce the dependency on error but also reduce the number of ancilla qubits required. Besides, we apply generalized interpolated walks to stationary state preparation, which is a fundamental part of adiabatic quantum computing.
The above results only use a series of gradually increasing parameters, namely a degenerate Q since all single-marked vertex graphs have a unique target state. Actually, Q is a Markov chain and can construct the transition relationship between interpolated walks in the generalized interpolated walks. Different from the unique marked state in single-marked vertex graphs, there are so many marked state in the multi-marked vertex graph with M = {m 1 , m 2 , . . . , m x }, such as |m 1 , . . . , |m x , (|m 1 + |m 2 )/ √ 2, . . . , (|m 1 + · · · + |m x )/ √ x, etc. For the graph with multiple marked states, we cannot construct a determined interpolated walk with gradually increasing parameters. Then we can choose a part of marked states and design the corresponding interpolated walks P(s 1 ), P(s 2 ), . . . , P(s r ). Here Q denotes the transition matrix between these interpolated walks, and q ij is designed based on the speed ratio of P(s j ) and P(s i ) reaching the marked state. We think there is something interesting that deserves to discuss in the future.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).