Exploring the Search Space of Quantum Programs - CiteSeerX
Exploring the Search Space of Quantum Programs Andr´e Leier University of Dortmund Dept. of Computer Science Chair of Systems Analysis 44221 Dortmund, Germany [email protected]
Wolfgang Banzhaf Department of Computer Science Memorial University of Newfoundland St. John’s, NL, A1C 5S7 Canada [email protected]
Abstract- Here we present a first study of search spaces and fitness landscapes in the context of the evolution of quantum programs. We consider small instances of the Deutsch-Jozsa problem as a starting point for the exploration of search spaces of quantum algorithms and analyze the structure of mutation landscapes using autocorrelation functions and information measures for characterizing their behavior. The relationship between landscape characteristics and quantum algorithm evolution is useful for improving the efficiency of the search process.
puters is computationally very expensive. This conspired with huge search spaces on even the simplest problems to make evolutionary approaches very difficult to achieve breakthrough solutions in the development of new quantum algorithms. It is therefore absolutely essential to improve the efficiency of evolutionary search on those landscapes. It is known that landscape structures have a strong influence on the ability of evolutionary search to perform efficiently. Digital circuit evolution [25, 26] was sucessfully investigated in this regard which might give a hint as to the behavior of improved quantum circuit evolution. A first step toward achieving more insight into the relationship between landscape characteristics and quantum circuit evolution is to study small problem instances of a well-known QC problem, like the Deutsch-Jozsa problem [5, 6]. Only a very small number of qubits is needed for this problem and it is therefore well qualified for experimental landscape analysis. The paper is organized as follows: Section 2 reviews some essential basics of quantum computing, Section 3 briefly outlines previous work on automatic (GP-based) quantum circuit design, Section 4 explains the general Deutsch-Jozsa problem. In Section 5 the fitness landscape for this problem is defined in terms of its fitness function and evolutionary operators. Furthermore, the genotypic representation in our experiments is described. Besides, we report the distribution of fitness in the Deutsch-Jozsa problem. In Section 6 we briefly summarize two recent techniques for landscape analysis, correlation analysis by Weinberger [28] and information analysis by Vassilev [24]. Both are applied to landscapes of the Deutsch-Jozsa problem. Section 7 describes our experimental setup and presents results. Finally, we draw conclusions in Section 8 and suggest future work.
1 Introduction With Shor’s discovery of a polynomial-time quantum algorithms for prime factorization and discrete logarithm [17] in 1994 quantum computing raised the hope of computational power far beyond that of conventional computers. Unfortunately, this discovery was not followed by many others. Today there are still only a few better-than-classical quantum algorithms, including the quadratic speed-up quantum search algorithm by Grover [8], quantum algorithms solving Hidden Subgroup Problems [11], Hogg’s 1-SAT and highly constrained k-SAT quantum algorithms [13] as well as some quantum algorithms for number theoretical problems like solving Pell’s equation [10], estimating Gauss sums [23] or solving shift problems [22]. Quantum algorithms are highly non-intuitive. This makes manual quantum circuit design difficult and motivates the search for computer-aided or automatic design techniques. The use of genetic programming to evolve quantum circuits was pioneered in 1997 by Williams and Gray [29]. Since then, various other papers [7, 19, 18, 3, 21, 14] dealt with quantum computing as an application of genetic programming or genetic algorithms, respectively. The primary goal of most GP experiments described in this context was to demonstrate the feasibility of automatic quantum circuit design. Different GP schemes and representations of quantum algorithms were considered and tested on various problems. Simulation of quantum circuits on conventional com-
2 Quantum Computing Basics The basic unit of information is the qubit which, unlike a classical bit, can exist in a superposition of the two classical states 0 and 1, i. e. with a certain probability p, resp. 1 − p, the qubit is in state 0, resp. 1. In the same way an n-qubit quantum register can be in a superposition of its 2n
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Figure 1: Some basic unitary 1- and 2-qubit transformations: Hadamard-gate H, the phase-gate Ph [φ] and the rotation gates Rx [φ] and Ry [φ] with angle parameter φ, NOT and CNOT. classical states s0 = 00 . . . 00, s1 = 00 . . . 01, . . . , s2n −1 = 11 . . . 11, also called base states. The state of the quantum register is described by a 2n -dimensional complex vector (α0 , α1 , . . . , α2n −1 ), where αk is the amplitude correP n −1 sponding to the classical state sk and 2k=0 |αk |2 = 1 is the normalization condition. The probability for a quantum register to be in state sk is |αk |2 when measured in the computational basis. By measurement a superposition collapses to the measured state. The quantum circuit model of computation describes quantum algorithms (circuits) as a sequence of unitary – and therefore reversible – transformations, also called quantum gates, which are applied successively to an initialized quantum state. A quantum gate operating on n qubits is a 2n ×2n matrix U , with U † U = I. Simple one-qubit gates are the Hadamard-gate (H ), rotation-gates (Rx [φ], Ry [φ]), phaseshift gates (Ph [φ]) and NOT. An important (k + 1)-qubit gate is the control-NOT (Ck NOT ) which flips the targetqubit if the k control-qubits are 1. Some of these gates are illustrated in Figure 1. In order to be applicable to an n-qubit quantum computer (with a 2n -dimensional state vector) quantum gates operating on less than n qubits have to be adapted to higher dimensions. For example, let U be an arbitrary single-qubit gate applied to qubit q of an n-qubit register. Then the entire n-qubit transformation is composed of the tensor product I ⊗ . . . ⊗ I ⊗U ⊗ I ⊗ . . . ⊗ I | {z } | {z } n−(q+1)
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Thus, calculating the new quantum state requires 2n−1 matrix-vector-multiplications of the 2 × 2 matrix U . It is easy to see, that the costs of simulating a quantum circuit on a conventional computer grow exponentially with the number of qubits.
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Figure 2: Matrix implementing NAND on three bits. The rightmost bit is the output, which is flipped if at least one of the two input bits is 0. E. g. applying this matrix to state s2 = 010 by multiplying the matrix with the vector (0, 0, 1, 0, 0, 0, 0, 0) results in state s3 = 011. Similar to the universality property of classical gates, small sets of quantum gates are sufficient to compute any unitary transformation to arbitrary accuracy. For example, all single qubit gates and CNOT form a universal basis for quantum computation, just as H, CNOT, Ph [π/4] and Ph [π/2] do. Inputs to quantum algorithms are provided by certain unitary matrices, sometimes known as oracles. They may change from instance to instance of a given problem, while the “surrounding” quantum algorithm remains unchanged. In Deutsch’s quantum algorithm [5, 6, 4] oracle gates are permutation matrices computing Boolean functions. Let f be such a function. Then, x → (x, f (x)) is one-to-one and a unitary matrix Uf exists, mapping (x, b) to (x, b ⊕ f (x)). For example, the matrix shown in Figure 2 implements the Boolean function NAND of two input bits, which outputs 1 for all inputs except 11. To get a deeper insight into quantum computing and quantum algorithms the following references might be of interest: [9, 15, 12].
3 Previous Work in Automatic Quantum Circuit Design Williams and Gray in [29] focus on demonstrating a GPbased search heuristics more efficient than an exhaustive enumeration strategy. It finds a correct decomposition of a given unitary matrix U into a sequence of simple quantum gate operations. In contrast, however, to subsequent GP schemes for the evolution of quantum circuits, a unitary operator solving the given problem had to be known in advance. Extensive investigations concerning the evolution of quantum algorithms were done by Spector et al. [19, 18, 2, 3]. In [19] they presented three different GP schemes for quantum circuit evolution: the standard treebased GP and both stack-based and stackless linear genome
GP. These were applied to evolve algorithms for Deutsch’s two-bit problem, the scaling majority-on problem, the quantum four-item database search problem, and the two-bitAND-OR problem. Better-than-classical algorithms could be evolved for all but the scaling majority-on problem. In [14] a linear-tree GP scheme was successfully applied to evolve a scalable quantum algorithm for 1-SAT, analogous to Hogg’s algorithm.
4 The Deutsch-Jozsa Problem Given a function f : {0, 1}n → {0, 1} (as a black box) promised to be either constant, f (x) = c, ∀x ∈ {0, 1}n and c ∈ {0, 1}, or balanced, i. e. as the result of f 0 occurs as many times as 1, the Deutsch-Jozsa problem is to determine which of the two properties f has. For n = 1 the task is also known as Deutsch’s problem. In classical computing f has to be evaluated 2n−1 + 1 times in the worst case, in quantum computing a single application of Uf is sufficient [4]. Using the above described matrix representation of a Boolean function (cf. Fig. 2) n + 1 qubits are necessary to solve the problem on a quantum computer. The number of Boolean functions being either constant or 2n balanced amounts to 2 + 2n−1 . The general quantum algorithm solving the Deutsch-Jozsa problem is discussed in detail in [9, 15].
5 Search Spaces and Fitness Landscapes Roughly speaking, a fitness landscape [20, 27] in evolutionary computation is a search space – the space of all phenotypes, represented by their genotypes – extended by the corresponding genotypes’ fitness values as the “altitude” and (optionally) a neighborhood structure. The fitness (dependent on the problem) decisively influences the contour and surface of the landscape. The neighborhood of genotypes is determined by an evolutionary (genetic) operator, typically an elementary mutation. Mathematically, in finite (discrete) spaces a fitness landscape can be considered to be a graph whose vertices are genotypes including an assigned fitness value and whose edges are defined by the evolutionary operator, obeying the so-called ”one-operator, one-landscape”concept [27]. In the following, we briefly describe the genotypic representation, the fitness function and the genetic operator as they are used in our experiments to analyze the fitness landscapes for Deutsch-Jozsa problem instances. Because quantum circuits have a natural linear structure1 it seems obvious to consider a linear genotype representation for quantum circuit evolution. It should be mentioned, that genotype and phenotype correspond directly to each 1 Intermediate measurements providing a linear-tree structure [14] are disregarded.
other. The population in evolutionary search methods moves on the fitness landscape by means of one or more evolutionary operators. Here, we exclusively consider the mutation operator, consisting of random deletion, insertion and replacement of a single quantum gate2 . The mutation is randomly selected (equally distributed) from the set of all possible mutations. The fitness of a quantum program is measured according to its quality to identify the property of a given function f . To obtain fitness values a quantum circuit is evaluated for each Boolean function matrix Uf with f either constant or balanced. To that end the sequence of gates (matrices) including Uf is applied to the initial base state s0 , resulting in a final quantum state. Afterwards, the results corresponding to balanced functions are compared with the results corresponding to the two constant functions (fitness cases). On the basis of measurement probabilities for every base state the determinability of classification is quantified. Misclassifications are penalized. A higher number of quantum gates in a genotype leads to a slightly increased fitness value due to a size penalty. The fitness function is standardized and normalized. A proper quantum algorithm, solving the Deutsch-Jozsa problem, classifies all functions correctly. We note that the Deutsch-Jozsa problem can also be solved by quantum algorithms which return the answer on a predefined qubit. These quantum circuits form a small subset of all well-classifying circuits and the fitness landscapes seem to be much harder to search by GP. This hypothesis is based on some GP experiments we made for Deutsch-Jozsa with n = 3. Our experiments were conducted using the universal gate (type) set {H, NOT, CNOT, CCNOT, Rx [φ], Ry [φ], INP }. INP denotes the input gate (the input matrix Uf ) corresponding to a Boolean function f which is either constant or balanced. The resolution of angle parameters φ, 0 ≤ φ < 2π, was restricted to 4 bits allowing rotations as multiples of 1/8π. The real set of gates available to build quantum circuits is of course much larger and depends on the number of qubits q the circuit consists of: H, NOT, Rx [φ] and Ry [φ] can be applied to each of these qubits, the number of CNOT and CCNOT gates corresponds to the number of possibilities to choose 1 resp. 2 control qubits and one target qubit out of q qubits. INP, however, is applied to all q qubits simultaneously and has no further degree of freedom. The total size of the gate set for different numbers of qubits is shown in the following table: 2 Due to our experience, we dropped the crossover operator in order to improve the efficiency of the search. A detailed analysis, as it is described in this paper for the mutation operator, is still pending.
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Figure 3 shows a quantum circuit in linear GP representation solving Deutsch-Jozsa for n = 2. To get a first impression of the quantum program spaces for instances of the Deutsch-Jozsa problem it might be helpful to take a closer look at the distributions of fitness values. The histogram in Figure 4 shows the Deutsch-Jozsa program space for n = 3. It is based on Monte Carlo sampling of 1, 000, 000 programs of maximum length L for L ∈ {10, 15, 20, 25, 30}. For L = 20 the evaluated quantum circuits occupy just about 1.8 × 10−38 of the entire search space (5, 58×1043). Fitness (deducting possible size penalties) does not exceed 0.7. It goes without saying that the search space increases with L exponentially.
6 Landscape Analysis A landscape can be characterized by its ruggedness and neutrality. Ruggedness refers to the average correlation between neighboring genotypes, neutrality refers to the size of flat landscape areas [16]. Since the study of landscape structures helps to understand the ability of evolutionary algorithms to perform efficiently, techniques for revealing these characteristics are important. Rugged and neutral fitness landscapes are usually considered to be difficult to evolve. The structure of fitness landscapes generated by evolving quantum circuits for the Deutsch-Jozsa problem was analyzed using two approaches: the autocorrelation analysis by Weinberger [28] and the information analysis by Vassilev et al. [24, 25, 27]. Both techniques are based on time series {ft }nt=0 obtained by random walks on the landscape. ft is the fitness value of the genotype reached after t steps from
Figure 4: Proportion of quantum circuits in fitness intervals from [0.0, 0.1[ to [0.6, 0.7] in the Deutsch-Jozsa program space for n = 3 (4 qubits). The histogram is based on Monte Carlo sampling of 1, 000, 000 programs of maximum length L for every L ∈ {10, 15, 20, 25, 30}. Note the logarithmic scaling on the quantity axis. the starting point. The autocorrelation function of a time series is defined by E[ft ft+s ] − E[ft ]E[ft+s ] ρ(s) = V [ft ] where E[ft ] is the expectation value and V [ft ] the variance. It is a measure for the correlation between points on the landscape that are separated by s steps, i. e. s applications of 1 the mutation operator. The correlation length τ = − ln ρ(1) is an important measure for the ruggedness of a landscape. Information analysis is used to investigate the structure in more detail, especially features related to flat areas (fitness plains), isolated points (locally isolated maxima or minima in the search space), and landscape structures having neither characteristic. The information measures being considered in this paper are • information content H(), to estimate the diversity of landscape shapes, • partial information content M (), to measure the modality of the landscape path, The information content is obtained by transforming the time series {ft }nt=0 into a string S() = s1 s2 . . . sn of symbols si ∈ {¯ 1, 0, 1}, where 1, if fi−1 − fi < − ¯ si = 1, if fi−1 − fi > 0, otherwise
7 Results For our analysis we performed random walks on problemspecific mutation landscapes for quantum circuits from 2 up to 4 qubits (n = 1 . . . 3) and maximum circuit sizes L = 10, 15, 20, 25, 30. Each random walk started with a random genotype and continued successively by random (one-point) mutation from one genotype to a neighbor. For each landscape, defined by L and n, 100 walks were performed, each of which consisting of 100, 000 steps. Both methods, autocorrelation analysis and information analysis, were applied to the same time series. The averaged autocorrelation functions of random walks for L = 10, 20, 30 are depicted in Figure 5. They show that correlations of (one-point) mutation landscapes are extremely low compared, for instances, to the autocorrelation of mutation landscapes for digital circuit evolution [25]. For increasing n and circuit sizes L (for each n) the autocorrelation increases only slightly. From these observations we conclude that the fitness landscape is rather rugged and seems to become only slightly smoother for larger n and L.
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for ∈ [0, l] where l is the maximum difference between two fitness values (the highest elevation in the fitness landscape). Then, Pthe information content of the time series is H() = − p6=q P[pq] log6 P[pq] , where P[pq] = n[pq] /n and n[pq] is the number of occurrences of pq, p, q ∈ {¯ 1, 0, 1}, in S(). Thus, P[pq] is the share of pq-blocks in S(). Increasing is like zooming out from the landscape, from a closer inspection to a rough overview. The partial information content is determined from S() by counting the number of slopes µ() in the landscape path which is equivalent to the number of changes in the fitness trend (plus 1). To evaluate µ() consider the string S(). Remove all symbols 0 and replace all sequences (runs) of equal symbols by a single symbol. The resulting string S 0 () is of the form 1¯ 11¯ 11¯ 1 . . . or ¯ 11¯ 11¯ 11 . . . and µ() is 0 just the length of S (). Thus, µ() indicates the modality of the landscape path. With it, the partial information content is given by M () = µ()/n. For example, for S() = 01¯ 11¯ 111100¯ 1¯ 101¯ 10 it is µ() = 8 and M = 0.5 0 ¯ ¯ ¯ since S () = 1111111¯ 1. The number of optima in the landscape path is b 12 µ()c. Another information measure is the density-basin information h(), used to estimate the variety of flat and smooth P areas. It is also based on the string S(): h() = − p∈{¯1,0,1} P[pp] log3 P[pp] . But instead of using h() we use the neutrality measures P[0] , P[00] , P[000] . . . directly which seems to be more expressive for this problem. P [0 . . . 0] is the frequency of (potentially overlapping) blocks 0 . . . 0 in S(0). For further information on the information landscape analysis it is referred to [27].
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Figure 5: Autocorrelation function of landscapes for Deutsch-Jozsa problems with n = 1 . . . 3 and L = 10, 15, 20. Note the logarithmic scaling. The standard deviation is < 0.009 considering ρ(s) averaged over 100 walks. The correlation lengths are given in Table 1. They reveal that beyond 2 steps most of the points on the landscape path become almost uncorrelated. τ n=1 n=2 n=3
L = 10 1.614 1.887 2.187
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Table 1: Correlation lengths. The information characteristics H() and M () are depicted in Figure 6. In order to achieve a better presentation we plotted the information measures for n = 2 and n = 3 in separate diagrams. Since the plots for n = 1 do not contribute to further conclusions they are not shown. For calculating the information characteristics the value of is varied with a stepsize of 0.001 between 0 and 1. figure In Figure 6 we can see that the averaged information content H(0) and the averaged partial information content M (0) increase as L increases. That is, the landscapes seem to become more rugged (in the sense that the frequencies of different blocks pq with p 6= q become more equal) and the number of local optima increases for higher values of L. Comparing the plots for n = 2 and n = 3 (for the same L) H(0) and M (0) decrease insignificantly. The difference between highest and lowest value of H(0) and M (0) for different values of L exceeds 0.1 and 0.14 respectively. That indicates different profiles of landscapes for larger genotypes. For every L (n = 2, 3) the value of H(0) is much larger than log6 2, which is in turn an indication for some flat land-
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Acknowledgement This work is supported by a grant from the Deutsche Forschungsgemeinschaft (DFG) under GK 726. We thank M. Emmerich, C. Richter and R. Stadelhofer for numerous discussions and helpful comments.
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The results confirm our experience from numerous GP evolutionary runs: Evolution of quantum circuits is very difficult, not only because of the expensive simulation on classical computers. The genotypic representation (the choice of a gate set) and the evolutionary operators seem to prevent efficient evolutionary search. Thus, it should be worth to examine how landscapes vary under different sets of quantum gates. Another approach might be to use novel evolutionary search operators. The standard one-point mutation operator could then be dropped. Exchanging gates at the same position in a quantum circuit has been found to be cause enormous variation in fitness.
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8 Conclusions and Outlook
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[15] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. [16] C. Reidys and P. Stadler, Neutrality in fitness landscapes, Tech. Rep. 98-10-089, Santa Fe Institute, 1998. [17] P. Shor, Algorithms for quantum computation: discrete logarithm and factoring, in Proceedings of the 35th Annual IEEE Symp. on Foundations of Computer Science (FOCS), IEEE, ed., pp. 124–134, IEEE Computer Society Press, Silver Spring, MD, USA, Nov. 1994. [18] L. Spector, H. Barnum, H. Bernstein, and N. Swamy, Finding a better-than-classical quantum AND/OR algorithm using genetic programming, in Angeline et al. [1], pp. 2239–2246.
[25] V. Vassilev, J. Miller, and T. Fogarty, Digital circuit evolution and fitness landscapes, in Angeline et al. [1], pp. 1299–1306. [26] V. Vassilev, J. Miller, and T. Fogarty, Digital circuit evolution: The ruggedness and neutrality of two-bit multiplier landscapes, in Proceedings of the IEE Half-Day Colloquium on Evolutionary Hardware Systems, D. Harvest, ed., pp. 6/1–6/4, IEE Press, London, 1999. [27] V. Vassilev, J. Miller, and T. Fogarty, Information characteristics and the structure of landscapes, Evolutionary Computation, 8 (1), pp. 31–60, 2000. [28] E. Weinberger, Correlated and uncorrelated fitness landscapes and how to tell the difference, Biologiacl Cybernetics, 63, pp. 325–336, 1990. [29] C. Williams and A. Gray, Automated Design of Quantum Circuits, in Explorations in Quantum Computing, C. Williams and S. Clearwater, eds., pp. 113–125, Springer, New York, 1997.
Wolfgang Banzhaf Department of Computer Science Memorial University of Newfoundland St. John’s, NL, A1C 5S7 Canada [email protected]
Abstract- Here we present a first study of search spaces and fitness landscapes in the context of the evolution of quantum programs. We consider small instances of the Deutsch-Jozsa problem as a starting point for the exploration of search spaces of quantum algorithms and analyze the structure of mutation landscapes using autocorrelation functions and information measures for characterizing their behavior. The relationship between landscape characteristics and quantum algorithm evolution is useful for improving the efficiency of the search process.
puters is computationally very expensive. This conspired with huge search spaces on even the simplest problems to make evolutionary approaches very difficult to achieve breakthrough solutions in the development of new quantum algorithms. It is therefore absolutely essential to improve the efficiency of evolutionary search on those landscapes. It is known that landscape structures have a strong influence on the ability of evolutionary search to perform efficiently. Digital circuit evolution [25, 26] was sucessfully investigated in this regard which might give a hint as to the behavior of improved quantum circuit evolution. A first step toward achieving more insight into the relationship between landscape characteristics and quantum circuit evolution is to study small problem instances of a well-known QC problem, like the Deutsch-Jozsa problem [5, 6]. Only a very small number of qubits is needed for this problem and it is therefore well qualified for experimental landscape analysis. The paper is organized as follows: Section 2 reviews some essential basics of quantum computing, Section 3 briefly outlines previous work on automatic (GP-based) quantum circuit design, Section 4 explains the general Deutsch-Jozsa problem. In Section 5 the fitness landscape for this problem is defined in terms of its fitness function and evolutionary operators. Furthermore, the genotypic representation in our experiments is described. Besides, we report the distribution of fitness in the Deutsch-Jozsa problem. In Section 6 we briefly summarize two recent techniques for landscape analysis, correlation analysis by Weinberger [28] and information analysis by Vassilev [24]. Both are applied to landscapes of the Deutsch-Jozsa problem. Section 7 describes our experimental setup and presents results. Finally, we draw conclusions in Section 8 and suggest future work.
1 Introduction With Shor’s discovery of a polynomial-time quantum algorithms for prime factorization and discrete logarithm [17] in 1994 quantum computing raised the hope of computational power far beyond that of conventional computers. Unfortunately, this discovery was not followed by many others. Today there are still only a few better-than-classical quantum algorithms, including the quadratic speed-up quantum search algorithm by Grover [8], quantum algorithms solving Hidden Subgroup Problems [11], Hogg’s 1-SAT and highly constrained k-SAT quantum algorithms [13] as well as some quantum algorithms for number theoretical problems like solving Pell’s equation [10], estimating Gauss sums [23] or solving shift problems [22]. Quantum algorithms are highly non-intuitive. This makes manual quantum circuit design difficult and motivates the search for computer-aided or automatic design techniques. The use of genetic programming to evolve quantum circuits was pioneered in 1997 by Williams and Gray [29]. Since then, various other papers [7, 19, 18, 3, 21, 14] dealt with quantum computing as an application of genetic programming or genetic algorithms, respectively. The primary goal of most GP experiments described in this context was to demonstrate the feasibility of automatic quantum circuit design. Different GP schemes and representations of quantum algorithms were considered and tested on various problems. Simulation of quantum circuits on conventional com-
2 Quantum Computing Basics The basic unit of information is the qubit which, unlike a classical bit, can exist in a superposition of the two classical states 0 and 1, i. e. with a certain probability p, resp. 1 − p, the qubit is in state 0, resp. 1. In the same way an n-qubit quantum register can be in a superposition of its 2n
NOT =
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Figure 1: Some basic unitary 1- and 2-qubit transformations: Hadamard-gate H, the phase-gate Ph [φ] and the rotation gates Rx [φ] and Ry [φ] with angle parameter φ, NOT and CNOT. classical states s0 = 00 . . . 00, s1 = 00 . . . 01, . . . , s2n −1 = 11 . . . 11, also called base states. The state of the quantum register is described by a 2n -dimensional complex vector (α0 , α1 , . . . , α2n −1 ), where αk is the amplitude correP n −1 sponding to the classical state sk and 2k=0 |αk |2 = 1 is the normalization condition. The probability for a quantum register to be in state sk is |αk |2 when measured in the computational basis. By measurement a superposition collapses to the measured state. The quantum circuit model of computation describes quantum algorithms (circuits) as a sequence of unitary – and therefore reversible – transformations, also called quantum gates, which are applied successively to an initialized quantum state. A quantum gate operating on n qubits is a 2n ×2n matrix U , with U † U = I. Simple one-qubit gates are the Hadamard-gate (H ), rotation-gates (Rx [φ], Ry [φ]), phaseshift gates (Ph [φ]) and NOT. An important (k + 1)-qubit gate is the control-NOT (Ck NOT ) which flips the targetqubit if the k control-qubits are 1. Some of these gates are illustrated in Figure 1. In order to be applicable to an n-qubit quantum computer (with a 2n -dimensional state vector) quantum gates operating on less than n qubits have to be adapted to higher dimensions. For example, let U be an arbitrary single-qubit gate applied to qubit q of an n-qubit register. Then the entire n-qubit transformation is composed of the tensor product I ⊗ . . . ⊗ I ⊗U ⊗ I ⊗ . . . ⊗ I | {z } | {z } n−(q+1)
q
Thus, calculating the new quantum state requires 2n−1 matrix-vector-multiplications of the 2 × 2 matrix U . It is easy to see, that the costs of simulating a quantum circuit on a conventional computer grow exponentially with the number of qubits.
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Figure 2: Matrix implementing NAND on three bits. The rightmost bit is the output, which is flipped if at least one of the two input bits is 0. E. g. applying this matrix to state s2 = 010 by multiplying the matrix with the vector (0, 0, 1, 0, 0, 0, 0, 0) results in state s3 = 011. Similar to the universality property of classical gates, small sets of quantum gates are sufficient to compute any unitary transformation to arbitrary accuracy. For example, all single qubit gates and CNOT form a universal basis for quantum computation, just as H, CNOT, Ph [π/4] and Ph [π/2] do. Inputs to quantum algorithms are provided by certain unitary matrices, sometimes known as oracles. They may change from instance to instance of a given problem, while the “surrounding” quantum algorithm remains unchanged. In Deutsch’s quantum algorithm [5, 6, 4] oracle gates are permutation matrices computing Boolean functions. Let f be such a function. Then, x → (x, f (x)) is one-to-one and a unitary matrix Uf exists, mapping (x, b) to (x, b ⊕ f (x)). For example, the matrix shown in Figure 2 implements the Boolean function NAND of two input bits, which outputs 1 for all inputs except 11. To get a deeper insight into quantum computing and quantum algorithms the following references might be of interest: [9, 15, 12].
3 Previous Work in Automatic Quantum Circuit Design Williams and Gray in [29] focus on demonstrating a GPbased search heuristics more efficient than an exhaustive enumeration strategy. It finds a correct decomposition of a given unitary matrix U into a sequence of simple quantum gate operations. In contrast, however, to subsequent GP schemes for the evolution of quantum circuits, a unitary operator solving the given problem had to be known in advance. Extensive investigations concerning the evolution of quantum algorithms were done by Spector et al. [19, 18, 2, 3]. In [19] they presented three different GP schemes for quantum circuit evolution: the standard treebased GP and both stack-based and stackless linear genome
GP. These were applied to evolve algorithms for Deutsch’s two-bit problem, the scaling majority-on problem, the quantum four-item database search problem, and the two-bitAND-OR problem. Better-than-classical algorithms could be evolved for all but the scaling majority-on problem. In [14] a linear-tree GP scheme was successfully applied to evolve a scalable quantum algorithm for 1-SAT, analogous to Hogg’s algorithm.
4 The Deutsch-Jozsa Problem Given a function f : {0, 1}n → {0, 1} (as a black box) promised to be either constant, f (x) = c, ∀x ∈ {0, 1}n and c ∈ {0, 1}, or balanced, i. e. as the result of f 0 occurs as many times as 1, the Deutsch-Jozsa problem is to determine which of the two properties f has. For n = 1 the task is also known as Deutsch’s problem. In classical computing f has to be evaluated 2n−1 + 1 times in the worst case, in quantum computing a single application of Uf is sufficient [4]. Using the above described matrix representation of a Boolean function (cf. Fig. 2) n + 1 qubits are necessary to solve the problem on a quantum computer. The number of Boolean functions being either constant or 2n balanced amounts to 2 + 2n−1 . The general quantum algorithm solving the Deutsch-Jozsa problem is discussed in detail in [9, 15].
5 Search Spaces and Fitness Landscapes Roughly speaking, a fitness landscape [20, 27] in evolutionary computation is a search space – the space of all phenotypes, represented by their genotypes – extended by the corresponding genotypes’ fitness values as the “altitude” and (optionally) a neighborhood structure. The fitness (dependent on the problem) decisively influences the contour and surface of the landscape. The neighborhood of genotypes is determined by an evolutionary (genetic) operator, typically an elementary mutation. Mathematically, in finite (discrete) spaces a fitness landscape can be considered to be a graph whose vertices are genotypes including an assigned fitness value and whose edges are defined by the evolutionary operator, obeying the so-called ”one-operator, one-landscape”concept [27]. In the following, we briefly describe the genotypic representation, the fitness function and the genetic operator as they are used in our experiments to analyze the fitness landscapes for Deutsch-Jozsa problem instances. Because quantum circuits have a natural linear structure1 it seems obvious to consider a linear genotype representation for quantum circuit evolution. It should be mentioned, that genotype and phenotype correspond directly to each 1 Intermediate measurements providing a linear-tree structure [14] are disregarded.
other. The population in evolutionary search methods moves on the fitness landscape by means of one or more evolutionary operators. Here, we exclusively consider the mutation operator, consisting of random deletion, insertion and replacement of a single quantum gate2 . The mutation is randomly selected (equally distributed) from the set of all possible mutations. The fitness of a quantum program is measured according to its quality to identify the property of a given function f . To obtain fitness values a quantum circuit is evaluated for each Boolean function matrix Uf with f either constant or balanced. To that end the sequence of gates (matrices) including Uf is applied to the initial base state s0 , resulting in a final quantum state. Afterwards, the results corresponding to balanced functions are compared with the results corresponding to the two constant functions (fitness cases). On the basis of measurement probabilities for every base state the determinability of classification is quantified. Misclassifications are penalized. A higher number of quantum gates in a genotype leads to a slightly increased fitness value due to a size penalty. The fitness function is standardized and normalized. A proper quantum algorithm, solving the Deutsch-Jozsa problem, classifies all functions correctly. We note that the Deutsch-Jozsa problem can also be solved by quantum algorithms which return the answer on a predefined qubit. These quantum circuits form a small subset of all well-classifying circuits and the fitness landscapes seem to be much harder to search by GP. This hypothesis is based on some GP experiments we made for Deutsch-Jozsa with n = 3. Our experiments were conducted using the universal gate (type) set {H, NOT, CNOT, CCNOT, Rx [φ], Ry [φ], INP }. INP denotes the input gate (the input matrix Uf ) corresponding to a Boolean function f which is either constant or balanced. The resolution of angle parameters φ, 0 ≤ φ < 2π, was restricted to 4 bits allowing rotations as multiples of 1/8π. The real set of gates available to build quantum circuits is of course much larger and depends on the number of qubits q the circuit consists of: H, NOT, Rx [φ] and Ry [φ] can be applied to each of these qubits, the number of CNOT and CCNOT gates corresponds to the number of possibilities to choose 1 resp. 2 control qubits and one target qubit out of q qubits. INP, however, is applied to all q qubits simultaneously and has no further degree of freedom. The total size of the gate set for different numbers of qubits is shown in the following table: 2 Due to our experience, we dropped the crossover operator in order to improve the efficiency of the search. A detailed analysis, as it is described in this paper for the mutation operator, is still pending.
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Figure 3: GP individual solving Deutsch-Jozsa for n = 2. f is constant, if qubit 1 and 2 are both measured 0, otherwise it is balanced. The first parameter after a gate type determines the qubit the gate is applied to, the second parameter is either an angle given by the number of multiples of 1/8π (with the real angle value in parenthesis) or a set of control qubits. Nr. of qubits Nr. of gates
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Figure 3 shows a quantum circuit in linear GP representation solving Deutsch-Jozsa for n = 2. To get a first impression of the quantum program spaces for instances of the Deutsch-Jozsa problem it might be helpful to take a closer look at the distributions of fitness values. The histogram in Figure 4 shows the Deutsch-Jozsa program space for n = 3. It is based on Monte Carlo sampling of 1, 000, 000 programs of maximum length L for L ∈ {10, 15, 20, 25, 30}. For L = 20 the evaluated quantum circuits occupy just about 1.8 × 10−38 of the entire search space (5, 58×1043). Fitness (deducting possible size penalties) does not exceed 0.7. It goes without saying that the search space increases with L exponentially.
6 Landscape Analysis A landscape can be characterized by its ruggedness and neutrality. Ruggedness refers to the average correlation between neighboring genotypes, neutrality refers to the size of flat landscape areas [16]. Since the study of landscape structures helps to understand the ability of evolutionary algorithms to perform efficiently, techniques for revealing these characteristics are important. Rugged and neutral fitness landscapes are usually considered to be difficult to evolve. The structure of fitness landscapes generated by evolving quantum circuits for the Deutsch-Jozsa problem was analyzed using two approaches: the autocorrelation analysis by Weinberger [28] and the information analysis by Vassilev et al. [24, 25, 27]. Both techniques are based on time series {ft }nt=0 obtained by random walks on the landscape. ft is the fitness value of the genotype reached after t steps from
Figure 4: Proportion of quantum circuits in fitness intervals from [0.0, 0.1[ to [0.6, 0.7] in the Deutsch-Jozsa program space for n = 3 (4 qubits). The histogram is based on Monte Carlo sampling of 1, 000, 000 programs of maximum length L for every L ∈ {10, 15, 20, 25, 30}. Note the logarithmic scaling on the quantity axis. the starting point. The autocorrelation function of a time series is defined by E[ft ft+s ] − E[ft ]E[ft+s ] ρ(s) = V [ft ] where E[ft ] is the expectation value and V [ft ] the variance. It is a measure for the correlation between points on the landscape that are separated by s steps, i. e. s applications of 1 the mutation operator. The correlation length τ = − ln ρ(1) is an important measure for the ruggedness of a landscape. Information analysis is used to investigate the structure in more detail, especially features related to flat areas (fitness plains), isolated points (locally isolated maxima or minima in the search space), and landscape structures having neither characteristic. The information measures being considered in this paper are • information content H(), to estimate the diversity of landscape shapes, • partial information content M (), to measure the modality of the landscape path, The information content is obtained by transforming the time series {ft }nt=0 into a string S() = s1 s2 . . . sn of symbols si ∈ {¯ 1, 0, 1}, where 1, if fi−1 − fi < − ¯ si = 1, if fi−1 − fi > 0, otherwise
7 Results For our analysis we performed random walks on problemspecific mutation landscapes for quantum circuits from 2 up to 4 qubits (n = 1 . . . 3) and maximum circuit sizes L = 10, 15, 20, 25, 30. Each random walk started with a random genotype and continued successively by random (one-point) mutation from one genotype to a neighbor. For each landscape, defined by L and n, 100 walks were performed, each of which consisting of 100, 000 steps. Both methods, autocorrelation analysis and information analysis, were applied to the same time series. The averaged autocorrelation functions of random walks for L = 10, 20, 30 are depicted in Figure 5. They show that correlations of (one-point) mutation landscapes are extremely low compared, for instances, to the autocorrelation of mutation landscapes for digital circuit evolution [25]. For increasing n and circuit sizes L (for each n) the autocorrelation increases only slightly. From these observations we conclude that the fitness landscape is rather rugged and seems to become only slightly smoother for larger n and L.
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for ∈ [0, l] where l is the maximum difference between two fitness values (the highest elevation in the fitness landscape). Then, Pthe information content of the time series is H() = − p6=q P[pq] log6 P[pq] , where P[pq] = n[pq] /n and n[pq] is the number of occurrences of pq, p, q ∈ {¯ 1, 0, 1}, in S(). Thus, P[pq] is the share of pq-blocks in S(). Increasing is like zooming out from the landscape, from a closer inspection to a rough overview. The partial information content is determined from S() by counting the number of slopes µ() in the landscape path which is equivalent to the number of changes in the fitness trend (plus 1). To evaluate µ() consider the string S(). Remove all symbols 0 and replace all sequences (runs) of equal symbols by a single symbol. The resulting string S 0 () is of the form 1¯ 11¯ 11¯ 1 . . . or ¯ 11¯ 11¯ 11 . . . and µ() is 0 just the length of S (). Thus, µ() indicates the modality of the landscape path. With it, the partial information content is given by M () = µ()/n. For example, for S() = 01¯ 11¯ 111100¯ 1¯ 101¯ 10 it is µ() = 8 and M = 0.5 0 ¯ ¯ ¯ since S () = 1111111¯ 1. The number of optima in the landscape path is b 12 µ()c. Another information measure is the density-basin information h(), used to estimate the variety of flat and smooth P areas. It is also based on the string S(): h() = − p∈{¯1,0,1} P[pp] log3 P[pp] . But instead of using h() we use the neutrality measures P[0] , P[00] , P[000] . . . directly which seems to be more expressive for this problem. P [0 . . . 0] is the frequency of (potentially overlapping) blocks 0 . . . 0 in S(0). For further information on the information landscape analysis it is referred to [27].
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Figure 5: Autocorrelation function of landscapes for Deutsch-Jozsa problems with n = 1 . . . 3 and L = 10, 15, 20. Note the logarithmic scaling. The standard deviation is < 0.009 considering ρ(s) averaged over 100 walks. The correlation lengths are given in Table 1. They reveal that beyond 2 steps most of the points on the landscape path become almost uncorrelated. τ n=1 n=2 n=3
L = 10 1.614 1.887 2.187
L = 15 1.659 1.927 2.257
L = 20 1.655 1.904 2.265
L = 25 1.627 1.860 2.255
L = 30 1.580 1.803 2.246
Table 1: Correlation lengths. The information characteristics H() and M () are depicted in Figure 6. In order to achieve a better presentation we plotted the information measures for n = 2 and n = 3 in separate diagrams. Since the plots for n = 1 do not contribute to further conclusions they are not shown. For calculating the information characteristics the value of is varied with a stepsize of 0.001 between 0 and 1. figure In Figure 6 we can see that the averaged information content H(0) and the averaged partial information content M (0) increase as L increases. That is, the landscapes seem to become more rugged (in the sense that the frequencies of different blocks pq with p 6= q become more equal) and the number of local optima increases for higher values of L. Comparing the plots for n = 2 and n = 3 (for the same L) H(0) and M (0) decrease insignificantly. The difference between highest and lowest value of H(0) and M (0) for different values of L exceeds 0.1 and 0.14 respectively. That indicates different profiles of landscapes for larger genotypes. For every L (n = 2, 3) the value of H(0) is much larger than log6 2, which is in turn an indication for some flat land-
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Acknowledgement This work is supported by a grant from the Deutsche Forschungsgemeinschaft (DFG) under GK 726. We thank M. Emmerich, C. Richter and R. Stadelhofer for numerous discussions and helpful comments.
Bibliography [1] P. Angeline, Z. Michalewicz, M. Schoenauer, X. Yao, and A. Zalzala, eds., Proceedings of the 1999 Congress on Evolutionary Computation, Washington DC, USA, IEEE Computer Society Press, Silver Spring, MD, USA, Jul. 1999. [2] H. Barnum, H. Bernstein, and L. Spector, Better-thanclassical circuits for OR and AND/OR found using genetic programming, Jul. 1999, LANL e-preprint quantph/9907056.
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[3] H. Barnum, H. Bernstein, and L. Spector, Quantum circuits for OR and AND of ORs, J. Phys. A: Math. Gen., 33 (45), pp. 8047–8057, Nov. 2000.
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The results confirm our experience from numerous GP evolutionary runs: Evolution of quantum circuits is very difficult, not only because of the expensive simulation on classical computers. The genotypic representation (the choice of a gate set) and the evolutionary operators seem to prevent efficient evolutionary search. Thus, it should be worth to examine how landscapes vary under different sets of quantum gates. Another approach might be to use novel evolutionary search operators. The standard one-point mutation operator could then be dropped. Exchanging gates at the same position in a quantum circuit has been found to be cause enormous variation in fitness.
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Figure 7: Neutrality measures P[0] , P[00] , . . .: Frequency of 0-blocks in S(0).
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8 Conclusions and Outlook
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This paper is a first step in the study of quantum circuit search landscapes. We studied mutation landscapes defined over small instances of the Deutsch-Jozsa problem. Our analysis shows that these fitness landscapes are characterized by high complexity and strong ruggedness making search very difficult, at least for larger quantum program sizes and/or problem instances.
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