Experiments testing the commutativity of finite-dimensional algebras with a quantum adiabatic algorithm

Computational and Mathematical MethodsVolume 1, Issue 1 e1009 RESEARCH ARTICLEFree Access Experiments testing the commutativity of finite-dimensional algebras with a quantum adiabatic algorithm Elías F. Combarro, Corresponding Author efernandezca@uniovi.es Department of Computer Sciences, University of Oviedo, Gijón, Spain Elías F. Combarro, Department of Computer Sciences, University of Oviedo, 33024 Gijón, Spain. Email: efernandezca@uniovi.esSearch for more papers by this authorJosé Ranilla, orcid.org/0000-0003-2941-3741 Department of Computer Sciences, University of Oviedo, Gijón, SpainSearch for more papers by this authorIgnacio F. Rúa, Department of Mathematics, University of Oviedo, Oviedo, SpainSearch for more papers by this author Elías F. Combarro, Corresponding Author efernandezca@uniovi.es Department of Computer Sciences, University of Oviedo, Gijón, Spain Elías F. Combarro, Department of Computer Sciences, University of Oviedo, 33024 Gijón, Spain. Email: efernandezca@uniovi.esSearch for more papers by this authorJosé Ranilla, orcid.org/0000-0003-2941-3741 Department of Computer Sciences, University of Oviedo, Gijón, SpainSearch for more papers by this authorIgnacio F. Rúa, Department of Mathematics, University of Oviedo, Oviedo, SpainSearch for more papers by this author First published: 24 January 2019 https://doi.org/10.1002/cmm4.1009Citations: 2 Present Address: Elías F. Combarro, Office 4.1, School of Geology, c/Jesús Arias de Velasco s/n, 33005 Oviedo, Spain AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract Determining whether a given algebra is commutative or not is important in the study of these algebraic objects in general and in the classification of semifields in particular. The best classical (ie, nonquantum) algorithm for this task has a running time that is of order O(n3), where n is the dimension of the algebra. To reduce this cost, in this paper, we study an approach to test the commutativity of a finite-dimensional algebra using quantum adiabatic computing. Previous quantum algorithms solving the same problem were based on Grover's quantum search. The algorithm is built from a quantum oracle for the multiplication constants of the algebra. Results of the experiments carried out on a quantum computer simulator, based on two different annealing schedules, are presented, showing that a quantum adiabatic algorithm for the problem can determine the commutativity of finite-dimensional algebras with one-side bounded error with a running time of order , achieving a quadratic speedup over the classical case. 1 INTRODUCTION The paradigm of quantum computing,1-4 which is based on a direct manipulation of atomic-scaled states of matter, has been theoretically shown to outperform classic computations in some cases. For instance, Grover5 and Shor6 algorithms, just to name the best well-known quantum algorithms, perform respectively quadratically and exponentially better than any other classic algorithm. Apart from them, other quantum algorithms exist,7 but it seems like there are not so many new alternatives that outperform classical techniques.8 That is the reason why it has been suggested that broadening the scope of problems in which the known quantum algorithms can be applied is an important research direction.9 In this sense, quantum algorithms for the study of problems in algebraic structures have been considered.10, 11 One of them is our recent study of commutativity of finite-dimensional algebras over an arbitrary field.12 It is based on the model of quantum gates that resembles the microdesign of classical algorithms from logical gates.13 Our algorithm was theoretically proved to be quadratically better than any classical algorithm and optimal among the quantum ones (when using the query model). In practice, however, no large-scale experiments have been made, as the current state of technology of general purpose quantum computers is rather limited,14 especially in the number of qubits of the available computers. Apart from the standard design of quantum algorithms from quantum gates, other approaches exist. Among them, one of the most popular is that of adiabatic algorithms, which are based on an evolution of the quantum states by a controlled series of Hamiltonians.15, 16 So far, this model has practically scaled better, with some quantum machines coping with up to some hundreds of qubits at the same time.17 In this context, we have developed a quantum algorithm for the study of commutativity of finite-dimensional algebras based on adiabatic techniques. The construction of the required Hamiltonians is based on the multiplication constant oracle introduced in our previous work and we are able to recover all the good properties of that previous algorithm in this new setting. The outline of this paper is as follows. Preliminaries on finite-dimensional algebras and the adiabatic quantum model are collected in Sections 2 and 3. Section 4 is devoted to present a general method to construct a Hamiltonian from a quantum oracle and to introduce our adiabatic algorithm based on this methodology. Finally, experiments carried out on an adiabatic quantum simulator and conclusions can be found in Section 5. 2 FINITE-DIMENSIONAL ALGEBRAS OVER A FIELD In this paper, K will be a field (it can be either finite, ie, a Galois field (see the work of Lidl and Niederreiter18) or infinite such as the real or complex number fields), and A will be a K-algebra, ie, a K-vector space equipped with a bilinear product.19 The algebra is commutative or associative if the multiplication satisfies the commutative or associative properties If the underlying K-vector space is finite dimensional, the K-algebra is called finite dimensional, and it is known as unital when the product has an identity element. Examples of unital algebras include division algebras (when any nonzero element has left and right multiplicative inverses), matrix rings over the field K, Lie and Jordan algebras (ie, K-algebras satisfying Lie or Jordan identities20) and finite semifields (ie, -finite-dimensional division algebras21). If A is an n-dimensional K-algebra ( ), and B = {x1,…,xn} is a K-basis of a A (ie, A = K < {x1,…,xn}>, and B is K-linearly independent), then there exists a unique set of constants such that This set of multiplication constants that completely determines the product in A is called multiplication table corresponding to A with respect to the basis B. It can be straightforwardly shown that A is a commutative algebra if and only if Mijk = Mjik, for all 1 ≤ i, j,k ≤ n. A quantum algorithm for deciding the commutativity of a finite-dimensional K-algebra A with basis B = {x1,…,xn} was introduced in our recent study.12 In that paper, the multiplication constants of the algebra A with respect to B are query-modeled input data. These multiplication constants (being elements of the field K) were given by an l-bit representation (when is a finite field with q elements, then l can be taken as , whereas when K is an infinite field, a standard numerical representation must be used). For technical reasons, the algebra A was embedded in an algebra that holds the same commutative character as A but whose dimension is slightly larger. In the present paper, we proceed in a similar way, but we benefit from the need of more conservative dimensions than in our previous work. As in the previous paper, an oracle made from the oracle will be used in the quantum computations. Then, here is an analogue of lemma 1 in our recent study12 that allows us to consider dimensions that are always a power of 2. Lemma 1.For any , take such that n ≤ 2m. Then, , and the -dimensional algebra K-algebra with the product given by the rule (a,λ)·(b,μ) = (ab,0) is commutative if and only A is. In particular, , and a query oracle for the multiplication constants of can be made from a query to the oracle OA for the multiplication constants of A. Proof.Obviously, the only way can be noncommutative is if there exist a and b in A such that a·b ≠ b·a in the original A, for on the new, added dimensions the multiplication is always 0. What is more, to recover a multiplication constant, we only need to check whether it corresponds to elements from A (in which case, we consult OA exactly once) or not (and then, we return 0). What is more, it is clear that this operation can be implemented in a quantum circuit (ie, with a unitary transformation) provided we have a quantum circuit for OA. In this context, we model the oracle in such a way that 3m index register qubits provide the encoding of the triple ijk while the multiplication constant is added to the l oracle qubits. From it, we build the problem oracle O f that models the binary function that marks with a one those multiplication constants Mijk, which are different from the corresponding Mjik, ie, those constants witnessing the noncommutativity of A. As such, an oracle is different when K has only two elements or it has more than two, as we will show in detail in Section 4. The main theorem in our recent study12 is the following. Theorem 1.There is a quantum decision algorithm on the gate model requiring queries to the oracle OA such that, on output NO, the noncommutativity of A is certified with certainty, while on output YES, the probability of A being noncommutative is a constant strictly smaller than 1. Moreover, it is query optimal among the quantum algorithms, in the sense that any other algorithm with bounded error probability for testing the commutativity of the finite-dimensional K-algebra A uses queries. Our objective in this paper is to construct a quantum adiabatic algorithm that recovers this quadratic speedup over the classical case while retaining the one-sided bounded error. 3 ADIABATIC QUANTUM ALGORITHMS Adiabatic quantum computation is a polynomially equivalent model to the standard gate model of quantum computation15, 22 that has been applied, for instance, to the simulation of quantum systems,23 to solving NP-complete problems,24 and to deep learning.25 Its theoretical foundation is the Quantum Adiabatic Theorem26 that states, roughly speaking, that, if a quantum system is prepared in an initial ground state of a Hamiltonian (ie, a state of minimal energy or, equivalently, an eigenvector associated to a minimal eigenvalue of the Hamiltonian), and the system is driven by a sequence of slightly changing Hamiltonians, then, with high probability, the final state will be also in the ground state of the last Hamiltonian.27 The idea is, then, to encode the information of the problem to solve by means of a Hamiltonian such that its ground states are the solutions to the problem. Usually, getting a system in a ground state of that Hamiltonian is not a simple task, so instead, we prepare a quantum system in an easy-to-get initial state and let it slowly evolve according to a time-dependent Hamiltonian that eventually reaches the desired, final Hamiltonian whose ground states are the solutions to the problem. Technically, let us assume that a quantum system evolves according to the Schrödinger equation in a time range 0 ≤ t ≤ T. If t(s) is a strictly increasing function from 0 to T as 0 ≤ s ≤ 1, then let us denote and . The Schrödinger equation can now be written as , for a certain function τ(s). Finally, if E0(s) and E1(s) are the first and second eigenvalues of the Hamiltonian , then let us denote the minimum spectral gap as . Theorem 2. (Adiabatic[28])Let |ψ0⟩ be the ground state of the Hamiltonian H(0) . If Δ > 0 and , where ‖·‖ is the matrix norm induced by the L2 metric, then, with high probability, |ψT⟩ is the ground state of H(T) . In practice, the computation procedure is implemented as follows. An objective real-valued function f on m variables, which is to be minimized, is considered An initial Hamiltonian HI is chosen, so that its ground state is easy to simulate in a register QI of m′ ≥ m qubits (some extra ancillary qubits might be necessary). A final Hamiltonian HF is built from f so that the ground state of QF is an eigenstate of HF having minimum eigenvalue, ie, it encodes a solution to the minimization problem of f. A changing Hamiltonian performing a gradual transition from HI to HF is created by the rule for all t ≤ T, and a certain function s(t) that increases from 0 to 1. By choosing T sufficiently large, the evolution of the initial state QI by that series of Hamiltonians can be brought arbitrarily close to QF, so that a final appropriate measurement provides the ground state of QF with high probability. It is clear that the computational complexity of an adiabatic quantum algorithm is largely determined by the minimum spectral gap that separates the ground state and the next excited state. The problem of determining the complexity of a quantum adiabatic process is, in general, hard, as stated in the work of Clark and Gaitan.29 Determining this gap in the limit of large problem size is currently an important open problem in adiabatic quantum computing […]. It is thus not possible to determine the computational complexity of adiabatic quantum algorithms in general, nor, consequently, of the specific adiabatic quantum algorithms presented in this paper. Fortunately, for our particular problem, we will be able to use some existing results about this gap, as we will mention in Section 5. Thus, in this paper, we apply the general framework of adiabatic quantum computation to our specific problem. The objective function to be minimized is f and, for that purpose, we create an appropriate final Hamiltonian from the oracle O f introduced above. How to achieve this in general will be explained in the next section. 4 FROM QUANTUM ORACLES TO HAMILTONIANS In this section, we state, for completeness, a general procedure to derive, from a given quantum oracle, a Hamiltonian that encodes, as its ground state, the solution to an optimization problem. Lemma 2.Let O be a quantum oracle that evaluates a binary function g on m ubits, ie, If M is the set of such that g(x) = 1 , then both O and are Hamiltonians whose ground state is generated by The corresponding eigenvalues are −1 and 0, respectively. Proof.It is enough to prove the lemma for O, as the second case can be straightforwardly deduced from it. First, notice that O2 = Id and that O† = O−1 because O is a unitary operator. Then, O† = O, the operator O is Hermitian, and therefore, it is a Hamiltonian.Furthermore, for all , we have and This means that , where E(λ) are subspaces of eigenvectors with eigenvalue λ ∈ { −1,1} Notice that this lemma is useful for it allows to transform the problem of searching states marked by an oracle from the quantum circuit model to the quantum adiabatic computational paradigm. In our case, for the problem of determining the commutativity of finite-dimensional algebras, we suppose we are given an oracle to access the multiplication constantes of the algebra. This oracle allows us to obtain (a quantum state representing) Mijk given (quantum states representing) i, j, and k as presented in Figure 1. Notice that we use l qubits to represent each constant, with l a number that will only depend on the size of the underlying field K (if K is infinite, an approximate representation must be adopted). Figure 1Open in figure viewerPowerPoint Oracle for the multiplication constants of the algebra From this circuit, we can then construct an oracle that will output 0 on those triples (i, j, j) for which Mijk = Mjik and 1 on those triples (i, j, j) for which Mijk ≠ Mjik; that is, it marks those indices that are witnesses of the noncommutativity of the algebra. If the underlying field is , the construction of this new oracle is straightforward: we just need to consult the constants Mijk and Mjik and compute their sum modulo 2. This can be accomplished with two consults to the oracle for the multiplicative constants and two swap operations (see Figure 2). Figure 2Open in figure viewerPowerPoint Problem oracle for fields with characteristic 2 For the general case, when K is not and more than one qubit is necessary to represent the underlying field, the construction is a bit more complex (see Figure 3) for we have to use ancilla qubits and we need to uncompute the values on those qubits to return their state to |0⟩. Figure 3Open in figure viewerPowerPoint Problem oracle for fields with characteristic different from 2 Notice, however, that the number of consults to the original multiplication oracle is, in both cases, a constant that does not depend on the dimension n of the algebra under study (namely, it is 2 if l = 1 and 4 if l > 1). We will use this construction in the next section in order to transform our original search problem into a quantum adiabatic optimization problem. 5 EXPERIMENTS AND RESULTS We have carried out experiments based on our previous theoretical considerations to determine if concrete -finite-dimensional algebras are commutative or not. Assuming that we have been given a multiplication constant oracle O A, we construct the corresponding final Hamiltonian HF = O f on 3m qubits using the results of Sections 2 and 4 ( ). Let us notice that, just like in our recent study,12 a standard phase-oracle technique allows to remove the last ancillary qubit in the construction of the oracle O f. Our choice for the initial Hamiltonian was where Notice that its ground state contains the initial state that was taken: |ψ0⟩ with eigenvalue equal to 0. In all the cases, the quantum adiabatic algorithm is used to search for a triple (i, j,k) such that Mijk ≠ Mjik. If such a triple is found, then we declare that the algebra is not commutative. In other case, we state that algebra is commutative. Explicitly, the algorithm is as follows. Notice that the only source of error of the algorithm comes from an answer in which we declare that the algebra is commutative, for we only say that is noncommutative we have found an explicit violation of the commutativity constrains. The practical details of the experiment are as follows. We have chosen dimensions n = 2,4,8, and 16. Furthermore, we have considered both global and local annealing schedules for the evolution of the Hamiltonian. In the first case, the linear function s(t) = t was chosen, whereas in the second case, the function s(t) was chosen according to the work of Roland and Cerf,30 that is, s verifying where and ϵ verifies The experiments were carried out in a standard computer using the QuTiP python package.31, 32 In the first experiment, we compare the behavior of global vs local evolution for algebras of dimensions n = 2,4,8, and 16. In these cases we assume that only one pair of multiplication constants (out of the n3 possible multiplication constants) witnesses the noncommutativity of the algebra. The probability of the adiabatic quantum algorithm finding one of those witnesses against the number of iterations made is plotted in Figures 4 through 7. Figure 4Open in figure viewerPowerPoint Probability of detecting the unique pair of noncommutative witnesses for -algebras of dimension n = 2 Figure 5Open in figure viewerPowerPoint Probability of detecting the unique pair of noncommutative witnesses for -algebras of dimension n = 4 Figure 6Open in figure viewerPowerPoint Probability of detecting the unique pair of noncommutative witnesses for -algebras of dimension n = 8 Figure 7Open in figure viewerPowerPoint Probability of detecting the unique pair of noncommutative witnesses for -algebras of dimension n = 16 As stated in the work of Roland and Cerf,30 global schedule yields a computation time of linear order, so there is no advantage of the method compared to a classical search. However, when the local schedule is considered, we have a quadratic speedup compared to classical search (so, the algorithm can be viewed as the adiabatic evolution version of the Grover's algorithm presented in our recent study12). Notice also that, in this case, the success probability exhibits an oscillatory pattern, as noticed in the works of Dalzell et al33 and Li et al.34 In the second experiment, we take a noncommutative algebra of dimension n = 8 with different numbers of multiplication constants witnessing its noncommutativity. Namely, we consider sets of 56, 112, 168, and 224 nonmatching pairs of multiplication constants out of the possible noncommutative witness pairs. The probability of finding one of those constants, when local search is carried out, is presented in Figure 8. Observe, once again, the oscillatory pattern in the success probability (even though it is predicted to converge to one as the number of iterations increases). Figure 8Open in figure viewerPowerPoint Probability of detecting a noncommutative witness pair (out of the 56, 112, 168, or 224) for a -algebra of dimension n = 8 Finally, we have studied the number of iterations required to find a unique noncommutative witness pair of multiplication constants with probabilities at least , , , and (for algebras of dimensions n = 2,4, and 8 with global and local schedules and also n = 16 with local schedule). As shown in Figures 9 and 10, the local annealing schedule yields better results than the global schedule: in the first case, the number of iterations increases with a factor of , whereas in the second case, the increasing factor is 23. Figure 9Open in figure viewerPowerPoint Number of iterations required to achieve a given probability of detecting the unique pair of noncommutative witnesses for -algebras of dimensions n = 2,4,8 with global schedule Figure 10Open in figure viewerPowerPoint Number of iterations required to achieve a given probability of detecting the unique pair of noncommutative witnesses for -algebras of dimensions n = 2,4,8,16 with local schedule 6 CONCLUSIONS AND FUTURE WORK In this paper, we have presented experiments testing the commutativity of finite-dimensional algebras with a Quantum Adiabatic Algorithm. This type of quantum model is a (polynomially equivalent) alternative to the standard quantum gate model that was considered for the same problem in our recent study.12 Experiments were carried out with both global and local annealing schedules, and it has been shown that, in the later case, Grover's order of complexity is recovered. This is technically relevant because the adiabatic quantum model has developed larger real models in practice so performance on a quantum computer with more number of qubits (and so, dealing with algebras of bigger dimensions) without sacrificing the theoretical complexity seems feasible. Furthermore, the new algorithm retains the property of having one side error: we always determine with certainty when the algebra under study is noncommutative and the only possible error is when we declare it to be commutative. In future works, we plan to explore the possibilities of applying quantum computing approaches (both with the quantum circuit model and with quantum adiabatic algorithms) to other problems related to the classification of finite-dimensional algebras in general and of semifields in particular, including, for instance, determining when an algebra is associative and finding all the matrices over the underlying field K that represent a semifield for a given dimension. ACKNOWLEDGEMENTS This work has been partially supported by the Ministry of Economy and Competitiveness (Ministerio de Economía y Competitividad) from Spain/FEDER under grant TEC2015-67387-C4-3-R, by the Ministry of Economy, Industry and Competitiveness (Ministerio de Economía, Industria y Competitividad) from Spain/FEDER under grant MTM2017-83506-C2-2-P, and by the Regional Ministry of the Principality of Asturias (Consejería de Empleo, Industria y Turismo del Principado de Asturias) under grants FC-GRUPIN-IDI/2018/000226 and FC-GRUPIN-IDI/2018/000193. Biographies Elías F. Combarro is an Associated Professor at Universidad de Oviedo (Spain). He holds degrees in both Mathematics (1997) and Computer Science (2002) and a PhD in Mathematics (2001). He is the author of more than thirty research papers in topics such as Computability Theory, Theory of Fuzzy Measures, Computational Classification of Semifields, and Text Categorization. Currently, his main research interest is Quantum Computing. José Ranilla is a Professor at Universidad de Oviedo (Spain). He holds degrees in Computer Science (1987 and 1993) and a PhD in Computer Science (1998). He is the author of more than 80 research papers in topics such as High-Performance and Parallel Computing, Computational Classification of Semifields, and Text Categorization. His current research interests also include Quantum Computing. Ignacio F. Rúa is an Associated Professor at Universidad de Oviedo (Spain). Graduated in Mathematics in 1999, he received his PhD in 2004. Research fellow of the Spanish "Juan de la Cierva" program at Universidad de Cantabria (Spain), from 2004 to 2007. Coauthor of some 30 research papers on nonassociative finite rings and their applications to coding theory and cryptography. His current research interests include also Computer Algebra and Quantum Computing. REFERENCES 1Feynman R. Simulating physics with computers. 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