Impact of input data alteration and modification of the algorithm
parameters on the efficiency of quantum programs
Principle investigator: Jaros law Miszczak
Institute of Theoretical and Applied Informatics, Polish Academy of Sciences January 30, 2020
1
Objectives
The goal of this project is to develop theoretical methods suitable for analysing the impact of quantum program alternation – input data modification or imprecise implementation of the algorithm – on the efficiency of quantum algorithms. Here quantum program is the sequence of quantum operations and the quantum representation of input data which are sent to the quantum processor. In some ceases we can consider quantum program alternation as a action of a malicious party and in this scenario we can understand it as an attack on quantum processor or quantum program.
The results of the project will enable the assessment of the applicability of quantum algo-rithms to different data structures, models of evolution, and the required accuracy of imple-mentation. Such assessment is needed to fully understand the computational power of quantum processors in the real-world scenarios and could provide essential milestones in the area of quan-tum computing. The proposed project is based on the novel approach which takes into account the structure of the data supplied as an input to quantum algorithms. Although the results obtained in the scope of the project will be of theoretical nature, they will have an impact on the development and deployment of quantum technologies.
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Significance
The primary motivation for research efforts in the area of quantum computing is the promise of achieving significant computational speed-up for algorithmic problems crucial in many areas of technology, including cryptography and optimization. Quantum information processing has been introduced as the new solution to many problems which appear in classical computer sci-ence. Two main applications were proposed and are still being developed: quantum protocols, which possess the level of security unachievable for protocols based on classical information, and quantum algorithms, which can break speed limitations of conventional computers [1]. In the second case, the best known algorithms are Shor’s algorithm, which can factorize numbers in polynomial time [2], and Grover’s algorithm for database search which achieves square root com-plexity. The progress in quantum computing motivated new developments in computer science and technology. In particular, the currently used cryptographic procedure could be rendered in-secure if large quantum computers are built [3]. The developments in quantum computing will have a significant impact on cryptography by posing a threat to the present cryptographic sys-tem. This motivates the current efforts in providing new standards for cryptographic procedures resistant to quantum algorithms [4]. The results of this project could be used to benchmark the
provide the methods for processing large data efficiently. For this reason, it is important to estimate to what degree the quantum algorithms can be applied to such data.
While the excitement in the area of quantum computing is fully justified by the new theoreti-cal developments, year by year scientists have discovered new limitations of quantum computing devices [5]. In particular, unitary operation decomposition provides a number of problems in-cluding the applications to hardware with fixed topology. Moreover, quantum algorithms have been proved to be sensitive to noise, which may impact the results of the computation. This re-sulted in the development of a new branch of quantum computing, namely the theory of quantum error-correcting codes. This aspect became even more critical when first commercial quantum computing systems became available. Furthermore, for quantum cryptographic protocols, hard-ware attacks, based on the security holes of conventional electronics, have been discovered [6]. This demonstrated that the theoretical security confirmed by the laws of physics in the ideal environment could lead to the creation of insecure protocols the real-world applications.
The main novelty of the presented project is the approach to the analysis of quantum algorithms based on the study of the input data. Using this approach we will characterize the applicability of quantum algorithms in real-world scenarios, taking into account that quantum programmes are exposed to malicious modifications, including those based on providing unsafe data.
Since quantum spatial search has been well analysed in the context of efficiency, we plan to use this family of algorithms as the starting point of the project. We will consider malicious input data modifications based on exceptional configuration and on the imprecise implementation of the algorithm. We will introduce the formal description of the resilience of the family of quantum algorithms based on quantum spatial search and provide the connection between the structure of the underlying network and the ability to perform an algorithmic attack on the algorithm. We will describe how the knowledge about the attack can be used to reduce the impact of the input data modifications and what is the dependency between the impact of the modification and the random graph model used to describe the data. Thus, our work provides the connection between the structure of the graph and the computational complexity of quantum algorithms.
We plan to generalize the results concerning quantum spatial search to other algorithms. In particular, we will provide results concerning the sensitivity of other quantum algorithms based on quantum walks in the similar situations. We will also consider the application of our results in the area of quantum simulators by exploiting the formalism of stochastic quantum walks, suitable for the description of general form of quantum evolution. We also intend to consider other families of quantum algorithms, for example, hidden-subgroup algorithm. However, at the moment of writing, we cannot specify the range of applicability of our approach to different algorithms.
The approach presented in this project aims at tackling the problem of attacking the quan-tum algorithms which is especially important taking into account that many quanquan-tum computing platforms are currently provided as a cloud service. At the same time, the understanding the limitations of quantum algorithms is crucial for a better understanding of the possibilities of quantum computing models. For those reasons, our proposition seems to be a novel and justified problem in quantum computer science. While in the scope of the project we will focus mostly on analysing quantum spatial search, the problem is much more general, and generalization to other types of quantum algorithms is of vital importance for the development of quantum technologies. We believe that our results will be important for the future development of quan-tum computing, as we propose an area that has not been analysed yet. They will enable the preparation of the existing quantum algorithms for working on real-world data.
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Figure 1: Example of modification based on exceptional configurations for Watts-Strogatz model. As the result of the malicious modification, the algorithms run-time complexity increased from nearly optimal Θ(n0.53) to Θ(n0.84).
3
Work plan
The proposed project will be divided into three work packages.
WP1 Quantum data modifications based on exceptional configurations
The goal of this WP is to study the modifications of input data on quantum spatial search (QSS) algorithms using exceptional configurations. We plan to consider different models of quantum walks. In particular, we plan to utilize the existing results concerning discrete models of quantum walks—Szegedy quantum walks (SzQW) and coined quantum walks (DCQW)—and utilize them to modify input data represented by various models of random graphs. We will also consider continuous-time quantum walks (CTQW) on closed system. We plan to provide the analysis of existence and form of exceptional configurations in this model. Again we plan to utilize the models of random graphs to describe real-world data.
The main result of this WP will be the development of mathematical tools suitable for reasoning and comparing the efficiency of quantum search algorithm using various models of quantum walks. In particular, we plan to use the formalism presented in Sec. 4. We will consider scenarios with different types of resources accessible by the attacker, including the access to the graph (add/remove edges), the access to the set of marked vertices (adding marked vertices), and the access to the parameters of the QSS typical for a chosen quantum walk.
In the scope of this WP, we also plan to continue the development of a simulation library enabling the numerical analysis of complexity attacks using the selected models of random graphs [7].
WP2 Decoherence-based modifications of quantum programs
The main objective of this WP is to study the impact of modifications realized by exploiting the effects related to the imprecise implementation of the algorithms. Such approach can be described using general form of quantum evolution and can be understood as the effects of decoherence. However, we will assume that the modifications are developed by taking into account the properties of quantum spatial search.
We will study the resources required to introduce an undesired effect. We will consider the limitation on the number of nodes available to the malicious party and the knowledge about the searched elements.
We also plan to consider which models are resistant to the methods based on decoherence. It is very unlikely that it is possible to counteract all such attacks. However, we hope to propose the models which will ensure security for some of the attacks at least for specific models of complex graphs.
WP3 Generalization to other families of algorithms
In the scope of this WP, we plan to investigate the possibility of generalizing the obtained results for other families of quantum algorithms. The most natural candidate are other algorithms based on quantum walks [8]. However, we also plan to tackle the problem of analysing complexity attacks on other types of quantum algorithms. In particular, we plan to focus on the algorithms based on hidden subgroup structure [2].
We plan to introduce the formal description of attacking efficiency, taking into account the construction of hidden-subgroup algorithms. We also plan to identify and exploit the weakness of quantum algorithms of this type with relation to the limitation of the architecture.For this purpose we plan to investigate the model of random geometric graphs for modeling the topology of available connections in the current quantum computing platforms.
The research results of this WP could be very important for the future developments of quantum computing and quantum technologies. However, at the moment of writing we are not able to foresee the possible outcome of this WP.
4
Methodology
Efficiency of quantum spatial search Quantum Spatial Search QSS is a tuple (Alg, t; G, S, θ), where Alg is a quantum algorithm searching for any vertex v ∈ S in time t, running on graph G, and parametrized by set of parameters θ.
Notation QSS → p denotes that QSS works with success probability p. Note that we do not define θ precisely, as it usually depends on chosen algorithm Alg. For example, in Alg = DCQW case, the parametrization θ consists of the set of coin operators used in each step for all vertices. For Alg = SzQW parametrization θ is the chosen stochastic operation P . In Alg = CTQW case, the parameters are the jumping rate γ and the label MG∈ {A, L} which describes graph
matrix chosen.
In order to compare different quantum spatial search algorithms we propose the following measure of efficiency.
Definition. Expected runtime TQSS of quantum spatial search QSS = (Alg, t; G, S, θ) → p is
defined as
TQSS :=
t
p. (1)
Note that the expected runtime is an increasing function of the graph order. Furthermore, one can show that the expected runtime is an expected time after ones which get the result using Bernoulli process. Such approach has been used in [9, 10], where the complexity was analysed. Definition. Attack on QSS is a function h such that
Note that the attack cannot change the evolution model and measure time of QSS. Still we will often restrict the function to change only some of the elements of QSS, for example attacks restricted to graph change imply that S = S0 and θ = θ0. Even in this case, we usually allow only small changes of the elements. We will use convention QSS −→ QSSh 0 as well.
If we allow QSS element to be changed, then we will say the element is susceptible. Other-wise, it is not susceptible. Note that by definition Alg and t are not susceptible.
Definition. Attack efficiency effh,QSS on QSS is defined as
effh,QSS := 1 −
TQSS
Th(QSS). (3)
We are interested in such h functions that effh,QSS ≥ 0. Furthermore, since t is common for
both QSS and h(QSS), so if QSS → p and h(QSS) → p0 we have effh,QSS = 1 − p0/p.
Suppose we have the following scenario: someone was trying to start algorithm QSS, but the attacker has changed it into QSS0. If the person has realized the algorithm was changed, he can steer measurement time t in order to maximize success probability. In this case, we need a stronger definition of the attack efficiency than the one provided earlier.
Definition. Suppose we have QSS = (Alg, t; G, S, θ) and h : QSS 7→ (Alg, t; G0, S0, θ0). Strong attack efficiency effh,QSS is defined as
effh,QSS := 1 −
T(Alg,t;G,S,θ)
maxτ ≥0T(Alg,τ ;G0,S0,θ0)
. (4)
Note that the efficiency here describes how well can we counteract against the undesired effect.
Exceptional configurations Recently, it has been shown that adding marked vertices can decrease the success probability of finding any of them [11, 12, 13]. Such configuration of marked vertices is called an exceptional configuration. Formally, the existence of an exceptional config-uration is shown in two steps: first, a special stationary state needs to be found, then a bound on probability is determined based on the stationary state. Recently, the class of connected sub-graphs having the stationary state has been described.
Theorem ([14]). Let G = (VG, EG) be an arbitrary graph. Let H ⊂ G be connected (VH1, VH2
)-bipartite subgraph. Then H contains a stationary state iff it satisfies X v∈V1 H degG(v) = X v∈V2 H degG(v), (5)
where degG is the degree in graph G. If H is not bipartite, then it contains a stationary state. In the case of CTQW and SzQW, there are the results demonstrating the existence of an exceptional configuration [15, 16]. While for the first model, it mostly affects the required jumping rate γ, in the second case we can observe the proper reduction of the success probability. Contrary to DCQW, the continuous and Szegedy models were not fully examined in the context of the exceptional configuration existence. Therefore, we plan to analyse and generalize the results using the methods similar to these already presented.
Acknowledgements
agree-References
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