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# On validity of quantum partial adiabatic search

*EPJ Quantum Technology*
**volume 11**, Article number: 47 (2024)

## Abstract

In this paper, we further verify the validity of the quantum partial adiabatic search algorithm which was initialized in the previous related works by revisiting its quantum circuit model. The main results got here are as follows. When considering implementing quantum partial adiabatic evolution on a quantum circuit, a correction is given for the time slice estimation for the first stage during this approximation in the previous related works, new evidence is provided for a time complexity cost \(O(\sqrt{N}/M)\) of quantum partial adiabatic algorithm is impossible, and the correct time complexity \(O(\sqrt{N/M})\) of it is emphasized once more according to its circuit correspondence, in which *N* is the total number of elements in the search problem of which *M* of them are the marked ones. The findings exposed are hopeful for revisiting quantum partial adiabatic evolution and its connection with the quantum circuit model.

## 1 Introduction

Quantum adiabatic evolution proposed by Farhi et al. [1, 2] was shown as a new approach to quantum computation, and had been proven to be equivalent to the traditional quantum circuit model [3, 4]. Nowadays, Quantum adiabatic computation(AQC) has been widely studied both in theoretical and practical sides. The article [5] is an excellent survey of AQC which covers topics that are essential to an understanding of the underlying principles of quantum adiabatic evolution, its algorithmic accomplishments and limitations, and its scope in the more general setting of computational complexity theory.

Quantum search is one of the dominating areas which demonstrates the superiority of quantum computation over classic computers. Especially, Grover had showed a clear picture for illustrating this [6]. However, when the quantum search was implemented directly in the framework of AQC, unlike Grover’s algorithmic quadratic speedup, this so called quantum global adiabatic search has no advantage over classic computation [1, 7]. Later, in Refs. [7, 8], the quantum local adiabatic evolution was proposed to remedy this problem, and the resulting search algorithm could thus show square-root speedup.

Since adiabatic evolution appeared, quantum adiabatic search had became an active area of research [9–17]. Especially, in [14] a particular class of quantum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state was analyzed in detail, and the minimum-energy gap which governed the time required for a successful evolution, was shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. Also, it was showed that such evolutions could exhibit a rapid crossover as the ground state changed abruptly near the transition point where the energy gap was minimum. Furthermore, a faster evolution could be obtained by performing a partial adiabatic evolution within a narrow interval around the transition point. After Tulsi’s work, Zhang et al. first considered the quantum search problem by partial adiabatic evolution explicitly [18, 19], and they found that the quantum partial adiabatic search algorithm could have a time complexity of \(T=O(\sqrt{N}/M)\) in which *M* out of the total size *N* of the problem were the target sets. So when there was single target element, the quantum partial adiabatic search could still exhibit a quadratic speedup over classic computation. Later, based on these results, in our previous work [20] we had proposed a quantum micro-local adiabatic search, and found that this algorithm asymptotically demonstrates the same performance as the one in [18], which implies the optimality of the quantum partial adiabatic search. In [21], we had explicitly showed that quantum partial adiabatic evolution could recover quantum global and local adiabatic computation respectively when appropriately tuning the evolving interval of it, which thus implies the flexibility of the former.

So it seemed that the research on quantum partial adiabatic search could come to a conclusion at that moment. However, soon after, Kay pointed out that an oversight in Tulsi’s paper which invalidates its proof [22], but the argument could be corrected by which the calculations in Tulsi’s paper were then sufficient to show that the scheme still works. Nevertheless, subsequent works [18–21, 23] couldn’t all be recovered in the same way.

In this paper, motivated by Kay’s work, we provide another way of verifying the validity of quantum partial adiabatic search in these works, namely, the quantum circuit viewpoint of the algorithm. From the early work [8, 24], we know that, when considering implementing the quantum local adiabatic evolution on the quantum circuit model, the time slices thus produced comes from estimating the difference of the resulting unitary operations from that of two supposed system Hamiltonians and implementing the whole unitary transformations by two elementary operations, respectively. And it was found that, these time slices are always consistent with each other, and they also match the time complexity of the quantum algorithm. Similar conclusion had also been reached in the case of quantum global adiabatic evolution [25]. But these shouldn’t come as a surprise just because of the equivalence of the two models of quantum computation. By the way, the recent novel outstanding results got in [26–29] emphasize once again the importance that we turn to the quantum gate model viewpoint of quantum computing for dealing with some problems.

In the work of [19], it can be checked that the time slice produced in the first stage of the approximation is of order O(1) and not consistent with that of the second stage, although the final total time slices coincides with the time complexity of the quantum partial adiabatic algorithm. Similar phenomenon has appeared in the work [30]. However, this is obviously not right and counterintuitive, which can be ultimately attributed to the equivalence between quantum adiabatic evolution and quantum circuit model. In this paper, we resolve this problem, show new evidence of invalidity of the quantum partial adiabatic searches in the previous related works mentioned above and argue their correct time complexity as being \(O(\sqrt{N/M})\) again, all by considering the quantum circuit model of the quantum adiabatic evolution. Especially, what we find here is that, if the quantum algorithm can have a time cost as \(O(\sqrt{N}/M)\), the resulting time slices doesn’t match it, which is impossible according to the results above concerning the approximation of quantum global and local adiabatic evolutions by quantum circuits. So our this new finding can be thought as another way of arguing the invalidation of the early related quantum partial adiabatic algorithms in addition to that of [22]. In contrast, the valid quantum partial adiabatic algorithm should provide a square root speedup over classic computers, just as that of quantum local adiabatic computation.

Finally, it should be noted that, although Kay also pointed out the result shown in [21] has the same flaw of calculation of the success probability in one round of partial adiabatic evolution, the evolving interval specified there was in fact acceptable, which could still lead to either the right time complexity of the quantum algorithm or the corresponding quantum circuit implementation.

## 2 Quantum partial adiabatic evolution revisited

For convenience, we first recall the quantum partial adiabatic evolution which was proposed by [14]. For this, consider the following system Hamiltonian

in which \(s=s(t)\in [0,1]\) is a monotone function of time. The two Hamiltonians in the above expression are constructed such that the quantum system is prepared in the ground state of \(H_{i}\) at \(t=0\) and the ground state of \(H_{f}\) is produced at \(t=T\), which corresponds to the solution of the problem of interest. In the quantum search problem, we consider a set of *N* states of which some unknown subset *S* are marked such that \(|S| = M\) with \(M\ll N\). Usually, when considering quantum adiabatic evolution of this problem, the initial and final Hamiltonians are designed respectively as

Obviously, if a quantum global adiabatic evolution is applied, the resulting algorithm has a time complexity \(T=O(N/M)\), which has no advantage over classic computation. And, if a quantum local adiabatic evolution is considered, the time complexity is \(T=O(\sqrt{N/M})\), which provides a square-root speedup.

Quantum partial adiabatic evolution was proposed in order to avoid the requirement of varying \(s(t)\) in time which is just case of quantum local adiabatic evolution, while still gaining the quadratic speedup. Specifically, for quantum search problem, the protocol involves the following three steps:

(1) Prepare the system in the state \(|\alpha \rangle \).

(2) Evolve the system Hamiltonian between two points \(s_{-}=\frac{1}{2}-\delta \) and \(s_{+}=\frac{1}{2}+\delta \) for some *δ*.

(3) Measure the system in the computational basis, and judge if the search was successful.

Repeat these steps until we find a marked state. In [18], *δ* in the two points above was specified as \(\delta =\frac{1}{2\sqrt{N}}\), which then led to an \(T=O(\sqrt{N}/M)\) time complexity for the quantum partial adiabatic search algorithm. And, when \(M=1\), the square-root speedup of the adiabatic evolution was recovered [19] again. In the followup work, we have considered that if in the (2) step above, a quantum local adiabatic evolution is executed instead, the resulting macro-local adiabatic evolution has no further algorithmic performance improvement, which implies the optimality of the algorithm in [18], if it were correct as it claimed. However, in [22] Kay pointed out that the results in these works were not correct and argued in detail why these algorithms couldn’t achieve such a performance. The origin is from the insufficiency of estimation of the successful probability of the algorithm in one run, which in turn comes from the insufficiency of bounding the overlap of the initial state with the ground state by a constant small than \(1/\sqrt{2}\). It was argued that if the parameter *δ* in the two points above is specified as \(\delta =\gamma \sqrt{\frac{M}{N-M}}\) in which *γ* is a constant, then the flaws in previous related works can be remedied. We should mention that, in fact our early works [21, 30] have already implied of this correct time points for the partial adiabatic evolution, even though the calculation for the successful probability is somewhat insufficient. By the way, as we know that quadratic speedup over classic computation being optimal for quantum algorithm [31], the \(T=O(\sqrt{N}/M)\) time complexity for a quantum partial adiabatic search is thus impossible, for which [30] has already implicitly evidenced. So our result that will be exposed next can be considered as a third way of arguing the invalidity of the quantum partial adiabatic search which has time complexity \(T=O(\sqrt{N}/M)\). Instead, it can also be seen that any correct quantum partial adiabatic algorithm should just have a time cost of \(O(\sqrt{N/M})\). But for clarity, we first show how we can obtain this correct time complexity explicitly.

From the early references such as [32], we know that the following sufficient condition for adiabatic approximation can be used:

in which we have assumed that \(\widetilde{H}(t)=H(s)\) is the reparameterized system Hamiltonian, and \(|E_{i}(t)\rangle \) and \(E_{i}(t)\) are the respective eigenvector and eigenvalues \(\widetilde{H}(t)\). In a quantum partial adiabatic search algorithm, the evolution paths should be specified as

in which \(T'\) is the one round time cost of the quantum partial adiabatic evolution. Now we have that

Applying the relation

the adiabatic condition in (3) can be rewritten as

in which

In general, we take the following form for simplicity,

which is sufficient enough for the one round time cost estimation. Therefore, the total time complexity of the quantum partial adiabatic algorithm can be bounded as follows,

in which *P* is success probability of a single run of the quantum partial adiabatic evolution.

From (1) and (2), it is easy to get the eigenvalue gap function of the system Hamiltonian \(H(s)\)

which then leads to that \(g_{min}=\sqrt{\frac{M}{N}}\).

By noting that \(s_{-}=\frac{1}{2}-\gamma \sqrt{\frac{M}{N-M}}\) and \(s_{+}=\frac{1}{2}+\gamma \sqrt{\frac{M}{N-M}}\), and using (9), we can obtain the one round time cost \(T'\geq 2\gamma \sqrt{\frac{N}{M}}\), in which we have used the assumption that \(N\gg M\). In the meanwhile, from [22] we know that the success probability *P* can be calculated out as

in which the constant *γ* is chosen such that \(\gamma \in [0, \sqrt{1-A^{2}}]\) for \(A=|\langle E_{0}(s_{-})|\alpha \rangle |>\frac{1}{\sqrt{2}}\) being satisfied. Therefore, by (10) we are finally led to the total time complexity of the quantum partial adiabatic search algorithm as \(T\geq \frac{\sqrt{\frac{N}{M}}(1+4\gamma ^{2})}{2\gamma}=O\big(\sqrt{ \frac{N}{M}}\big)\).

## 3 Circuit model of quantum partial adiabatic search

In this section, we first show in detail how to implement the correct quantum partial adiabatic search algorithm on a quantum circuit, during which one can find out that there was an error of the time slice estimation in the first stage of this process in the previous related works. We will resort to the following theorem for estimating the time slice thus produced in this stage, whose proof is similar to that in [24].

### Theorem 1

*Suppose that* \(H(t)\) *and* \(H'(t)\) *are two time*-*dependent Hamiltonians for* \(t^{-}\leq t\leq t^{+}\) *in which* \(t^{-}\) *and* \(t^{+}\) *denote the start and end time point of the quantum partial adiabatic evolution*, *and let* \(U(t)\) *and* \(U'(t)\) *be the respective unitary evolutions that they induce*. *If the difference between the Hamiltonians is limited by* \(|||H(t)-H'(t)|||_{2}\leq \delta (t)\) *for every* *t*, *in which for an operator* *M* *its* \(\ell _{2}\) *induced operator norm* \(|||\circ |||_{2}\) *is defined as* \(|||M|||_{2}:=\max _{||x||_{2}=1}||Mx||_{2}\), *then the distance between the induced transformations is bounded by* \(|||U(t^{+})-U'(t^{+})|||_{2}\leq \sqrt{2\int _{t^{-}}^{t^{+}}\delta (t)dt}\).

### Proof

Let \(|\Psi (t)\rangle \) and \(|\Psi '(t)\rangle \) be the respective quantum states which are induced by the Hamiltonians \(H(t)\) and \(H'(t)\) from the common initial state \(|\Psi (t^{-})\rangle =|\Psi '(t^{-})\rangle =|\Psi _{t^{-}}\rangle \). Thus, we can get the following equality

such that

Integrating the above inequality from \(t^{-}\) to \(t^{+}\), and noting that \(\langle \Psi (t^{-})|\Psi '(t^{-})\rangle =1\), we can get that

which then results the following inequality

Due to the arbitrage of \(|\Psi _{t^{-}}\rangle \), we can finally obtain that

The following inequality is easy to get

in which we have denoted the other Hamiltonian \(H'(s)\) as \(H'(s)=(1-s'(t))H_{i}+s'(t)H_{f}\), and used the fact that \(|||H_{i}-H_{f}|||_{2}=\sqrt{1-M/N}\). Let \(s'(t)\) be chosen such that \(s'(t)=s(t+\Delta T)\) where \(\Delta T=T/R\) with *R* the time slice, which then leads to that

in which we have assumed that \(s(t)=t/T\). It should be noted that, a quantum partial adiabatic evolution in fact can be considered as a special kind of quantum global adiabatic evolution where the evolving path is confined to a narrow interval. So unlike that in the previous work such as [25], in which the function of time \(s(t)\) in the evolving path was inferred precisely, which however led to the wrong time slice estimation of the first stage when considering approximating the adiabatic evolution by quantum circuit. In fact, here we instead adopt the function of time \(s(t)=t/T\), which causes no problem because of the fact that \(s(t^{-})\approx s(0)=0\) and \(s(t^{+})\approx s(1)=1\). And it will be more clear soon that due to this coarser choice of \(s(t)\), we can obtain the right estimation of time slice in this stage of the approximation. By the Theorem above, we have that

in which we have used that \(t^{+}-t^{-}=T'\approx T\) which is the one round of running time of the quantum partial adiabatic evolution. It is noted that here \(T\approx T'\) indeed can be satisfied as long as the success probability *P* is sufficiently high such that only one run of the quantum partial adiabatic evolution is required, which in turn enforces that the constant *γ* in the parameter *δ* of the two time points \(s_{-}\) and \(s_{+}\) is large enough. So from (20), we can easily obtain the time slice as \(R_{ad1}=O(\sqrt{N/M})\), in which we have used that the time complexity of the correct quantum partial adiabatic algorithm is \(T=O(\sqrt{N/M})\). So we have seen that this time slice is quite unlike that appeared in the previous related works [19, 21], which is given as \(O(1)\), and obviously causes confusion because of its violation of equivalence between quantum adiabatic evolution and quantum circuit model. By the way, it is easy to see that the time slice got in this stage is always consistent with time complexity of the quantum adiabatic algorithm, which should come as no surprise just because of the equivalence between those two models of quantum computation.

Next, we turn to the second stage of approximating quantum partial adiabatic evolution by quantum circuit, for which it deals with the problem of implementing the unitary transformations defined by

with elementary operations defined by

The Campbell-Baker-Hausdorff theorem [33] tells us how well we can approximate the unitary evolution in (21) by the one in (22). Therefore, the error estimation is given as follows,

which together with the fact \(|||[H_{i},H_{f}]|||_{2}=\sqrt{\frac{M}{N}}\sqrt{1-\frac{M}{N}}\) yields that

Also, it is known that the following inequality holds

which can be proved by induction. For \(R_{ad2}\) steps, it is known that \(U'(T)=\prod _{k} U_{k}^{\prime }\), which combined with the inequality above can give us

So \(R_{ad2}=O(\sqrt{N/M})\), which together with \(R_{ad1}=O(\sqrt{N/M})\), leads to the total time slices needed for the approximation above is \(R=O(\sqrt{N/M})\), which is consistent the time complexity \(T=O(\sqrt{N/M})\) of the algorithm.

If the time complexity of the quantum partial adiabatic search or macro-local adiabatic search can be \(T=O(\sqrt{N}/M)\), we turn to see what will happen when considering the quantum circuit model of the adiabatic evolution. Obviously, in the first stage, it is quite easy to see that \(R_{ad1}=O(\sqrt{N}/M)\) at this time from the previous discussion of this section. However, in the second stage, by (23), it is easy to get that

which further leads to that \(R_{ad2}=O(\frac{\sqrt{N/M}}{M})\) by using (25). So \(R_{ad2}\) doesn’t match the time complexity of the algorithm, which implies invalidity of the corresponding quantum partial adiabatic evolution.

In [25], for the circuit model of quantum global adiabatic evolution, we have adopted another form of the Campbell-Baker-Hausdorff theorem, which says that

Due to some similarity between the quantum global and partial adiabatic evolutions, we try to resort to the formula above for the time slice estimation in the second stage, since in the first stage the time slice estimation remains unchanged even if we should apply another theorem in [25] for it. By noting \(|||H_{i}H_{f}|||_{2}=M/N\) and using (25), we are finally led to that \(R_{ad2}=O(1)\), which obviously causes confusion and doesn’t match the time complexity of the algorithm, either. So we see the impossibility of the time complexity \(T=O(\sqrt{N}/M)\) for a quantum partial adiabatic algorithm once more.

On the other hand, we can rewrite (23) as follows

which together with (25) yields that

So by the equivalence between the quantum circuit model and quantum adiabatic evolution, which means that \(R_{ad2}=O(T)\), we finally get that the correct time complexity of the quantum partial adiabatic search algorithm should be \(T=O(\sqrt{N/M})\) as well. □

## 4 Conclusions and discussions

In this paper, we have delivered some quantum circuit model implementations of the quantum partial adiabatic evolutions which were proposed previously. Due to that there exists some flaw for the time slices estimation in the first stage of this procedure in the previous related works, we have shown how to correct it, and find that the result is consistent with the time complexity of the algorithm, in contrast to that of \(O(1)\) in these works, which is obviously not right and counterintuitive. Also, for the correct quantum partial adiabatic search algorithm, we find that the time slices produced in the two stages of this process matches each other, and both of them equals to the algorithmic complexity of the adiabatic evolution, which shouldn’t come as a surprise just because of the equivalence between the two models of quantum computation. However, when an \(O(\sqrt{N}/M)\) time complexity of quantum partial adiabatic search is considered for its circuit model, the resulting time slices in the second stage doesn’t match \(O(\sqrt{N}/M)\) through either of the two Campbell-Baker-Hausdorff theorems used in the literatures before, which thus implies incorrectness of this algorithm. Although [22] has already pointed out the right way of remedying the mistake made in these works, here we take the viewpoint of implementing adiabatic evolution by quantum circuit on this problem, which can also give a judgement on the correctness of the algorithm. On the other hand, if we always bear in mind that the quantum adiabatic evolution is equivalent to quantum circuit model in the computational power aspect, we also arrive at that the correct algorithm complexity of quantum partial adiabatic search should be \(O(\sqrt{N/M})\) instead of \(O(\sqrt{N}/M)\).

The results here are hoped to be useful for further understanding the quantum partial adiabatic search or even quantum partial adiabatic evolution as a paradigm of quantum computation, and its connection with the quantum circuit model. By providing a more accurate understanding of the time complexity and correcting previous errors in the related references, this paper may pave the way for more reliable and efficient implementations of quantum search algorithms. Future research can build on these findings to further optimize quantum partial adiabatic algorithms and explore their applications in various quantum computing tasks. While the main result here makes significant contributions to the understanding and correction of the quantum partial adiabatic search algorithm, there are several potential issues and areas that might not have been fully addressed. Below we just list some of them.

Firstly, here we have argued that the genuine time complexity of the quantum partial adiabatic search algorithm should be \(O(\sqrt{N/M})\) rather than \(O(\sqrt{N}/M)\), and corrected theoretical estimation of time slices produced in the first stage of implementing quantum partial adiabatic evolution by quantum circuit model. But we do not deeply explore the practical challenges of scaling the quantum partial adiabatic search algorithm. How these corrections affect real-world quantum systems with noise and decoherence might require further exploration.

Secondly, since at present quantum computing is entering the NISQ era, considering implementing the corrected quantum partial adiabatic search algorithm on current quantum hardware is vital important for us to understand the quantum algorithm in the practical side. And for this, we should make a detailed analysis of the resources such as qubits and gates required for the implementation, as we know that understanding the feasibility of the corrected algorithm on near-term quantum devices is crucial. Also, experimental validation on actual quantum hardware or detailed simulations could further strengthen the findings here, because real-world experiments can uncover unforeseen challenges and confirm the practical validity of the corrections.

Thirdly, also in the paper we haven’t been addressed the impact of quantum errors and the need for fault-tolerant designs in the implementation of the corrected quantum partial adiabatic algorithm. As shown in the early works such as [34–37], the quantum adiabatic evolutions are inherently robustness against errors. Qualitatively speaking, the quantum partial adiabatic search could inherit this virtue in principle just because of that it is both of a special kind of quantum global and local adiabatic evolution. However, quantitative research on how robust it is again errors in real quantum circuits deserves deep exploitation in the future.

Fourthly, in this paper we only focus on the quantum partial adiabatic search itself to clarify its validity. Though the result here show that the quantum partial adiabatic search should have the same time complexity as Grover’s algorithm, a more concise time cost comparison between them may provide more insights into the relative algorithmic performance and practical advantages or disadvantages of the former quantum algorithm. As we have already seen from Sect. 2 of this paper or that from [22], the precise time assumption of either the original quantum partial adiabatic search or its optimized version can be calculated out. From [38], we know this is also true for Grover’s algorithm. So the comparison work can be done rather straightforwardly. But in the current paper, we leave it as a near future work, since we think that it will also supplement the experimental findings of these quantum algorithms.

Last but not least, exploration of the broader implications of the corrected time complexity for the quantum partial adiabatic algorithms or applications beyond search problems might add more context to the significance of the findings in this paper. Perhaps we can try to apply the quantum partial adiabatic evolution to solve some prominent combinatorial optimization problems as a first step toward this goal.

## Data Availability

No datasets were generated or analysed during the current study.

## Code availability

Not applicable.

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## Acknowledgements

We sincerely thank the anonymous reviewers for their insightful feedback and constructive criticism on our paper. Their expertise and suggestions have undoubtedly improved the quality and clarity of our work.

## Funding

The first author’s work in this paper is supported by the General Program of Educational Commission of Anhui Province of China under Grant No. K120455011, and the Research Start-up Funds of Anhui University under Grant No. M080255003.

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Jie Sun wrote the main manuscript text and reviewed the manuscript. Dunbo Cai analyzed the mathematical result and reviewed the manuscript. Songfeng Lu validated the mathematical proof and reviewed the manuscript. Ling Qian and Runqing Zhang edited and reviewed the manuscript.

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Sun, J., Cai, D., Lu, S. *et al.* On validity of quantum partial adiabatic search.
*EPJ Quantum Technol.* **11**, 47 (2024). https://doi.org/10.1140/epjqt/s40507-024-00258-6

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DOI: https://doi.org/10.1140/epjqt/s40507-024-00258-6