Gene Expression Programming for Quantum Computing

Gene Expression Programming for Quantum Computing GONZALO ALVAREZ, Oak Ridge National Laboratory, USA RYAN BENNINK, Oak Ridge National Laboratory, USA STEPHAN IRLE, Oak Ridge National Laboratory, USA JACEK JAKOWSKI, Oak Ridge National Laboratory, USA We introduce QuantumGEP, a scientific computer program that uses gene expression programming (GEP) to find a quantum circuit arXiv:2303.08203v1 [quant-ph] 14 Mar 2023 that either (i) maps a given set of input states to a given set of output states, or (ii) transforms a fixed initial state to minimize a given physical quantity of the output state. QuantumGEP is a driver program that uses evendim, a generic computational engine for GEP, both of which are free and open source. We apply QuantumGEP as a powerful solver for MaxCut in graphs, and for condensed matter quantum many-body Hamiltonians. Additional Key Words and Phrases: genetic algorithms, gene expression programming, quantum computing, condensed matter, quantum chemistry ACM Reference Format: Gonzalo Alvarez, Ryan Bennink, Stephan Irle, and Jacek Jakowski. 2022. Gene Expression Programming for Quantum Computing. 1, 1 (March 2022), 10 pages. https://doi.org/XXXXXXX.XXXXXXX 1 INTRODUCTION One of the fundamental problems in quantum computing is to find a quantum circuit that, when applied to a prepared initial state, yields the ground state of a quantum many-body Hamiltonian. This is a conventional yet difficult problem that can in principle be solved by machine learning. We have chosen gene expression programming (GEP) [7, 8] as a tool to generate quantum circuits for applications. Only recently has interest in automatically generated quantum circuits increased [10], so that our goal of introducing a free and open source tool for quantum circuit generation appears timely. Other solutions to the same or similar problems have used the quantum approximate optimization algorithm (QAOA), [4, 6, 15] a variational quantum-classical algorithm for combinatorial optimization problems. An iterative version of QAOA has recently been developed[22] so that it is problem-tailored; it can also be adapted to specific hardware constraints. In our approach using GEP, a set of initially random quantum circuits (“population”) is evolved by repeatedly introducing variations of the existing circuits (“mutations”) and retaining only those individuals that perform best at the prescribed computational task (have the highest “fitness”). After sufficiently many iterations (“generations”), the population should consist mostly of circuits that perform well at the prescribed task. To accomplish the mutations, each individual is specified by a string of symbols called a genome. Mutations are small random changes to these strings of symbols, yielding generally new and different circuits. The fitness of an individual is evaluated via simulation, but in principle could be evaluated efficiently by executing the circuit on a quantum computer. Authors’ addresses: Gonzalo Alvarez, Oak Ridge National Laboratory, Oak Ridge, USA; Ryan Bennink, Oak Ridge National Laboratory, Oak Ridge, USA; Stephan Irle, Oak Ridge National Laboratory, Oak Ridge, USA; Jacek Jakowski, Oak Ridge National Laboratory, Oak Ridge, USA. 2022. Manuscript submitted to ACM Manuscript submitted to ACM 1 2 Gonzalo Alvarez, Ryan Bennink, Stephan Irle, and Jacek Jakowski Q + + b ∗ a Q ∗ − a − a b c d a √︁ Fig. 1. (a) AST for 𝑎 ∗ 𝑏 + (𝑐 − 𝑑), which has the string representation Q+*-abcd, operator (or primitives in GEP) in {Q, +, −} and leaves (shown shaded in the figure) in {𝑎, 𝑏, 𝑐, 𝑑 }. (b) The set of operators here is as in (a), with inputs 𝑎 and 𝑏. The twenty character AST string +b*aQa-ababbabbbabab has a coding region (the first eight characters) that is shown pictorially as an AST. The last twelve characters are non coding [8]. This work introduces QuantumGEP, a scientific computer program that uses GEP to find a quantum circuit that either (i) maps a given set of input states to a given set of output states, or (ii) transforms a fixed initial state to minimize a given physical quantity of the output state. Both tasks are common in quantum mechanics. Training a quantum computer to learn a polynomial [21] or perform a quantum Fourier transform are examples of the first kind of task. Finding the ground state energy of a molecule [12] or solving a combinatoric optimization problems [11, 14] are examples of the second kind of task. QuantumGEP is a driver program that uses evendim, a generic computational engine for GEP that we developed from scratch; both QuantumGEP and evendim are free and open source. Section 2 discusses the representation of quantum circuits by GEP strings, and the two algorithms used, one for arbitrary function representation, and one for ground state of a Hamiltonian. We discuss the GEP features currently available in 2.4. Section 3 discusses two applications, one to the MaxCut problem in graphs, and one to ground state properties of many-body Hamiltonians as they appear in condensed matter and quantum chemistry. Section 5 presents a summary and outlook for future work. 2 THEORY AND IMPLEMENTATION 2.1 Gene Expression Programming Any computer program [1] can be written in abstract syntax tree (AST) form; Figure 1(a) shows an example. To understand this figure, one first needs to consider what primitives or operators to include, which depends on the problem. In the example, we have chosen from the set {𝑄, ∗, +, −}, where 𝑄 is the square-root operator, and the other symbols stand for their usual algebraic meaning. The operator or primitive 𝑄 is unary, that is, it takes only one argument whereas the other operators are binary requiring two arguments each. In the example, the leaves, or inputs, are taken from the set {𝑎, 𝑏, 𝑐, 𝑑 } and can be considered inputs or numbers. We can represent any AST not just pictorially, but also in string form: we do this by writing down the primitives and leaves in breadth-first order. The AST shown in Figure 1(a) has thus the string representation 𝑄 + ∗ − 𝑎𝑏𝑐𝑑. In the process of our artificial intelligence software finding a suitable mathematical function, we can think of an AST as an individual life form; many ASTs form a population of life forms. More specifically, we think of the AST as the genome and body of an individual; when we have a population of genomes they reproduce and create a new generation or population from the previous one. This is done by combining and mutating genes, and selecting the survivors that come closest to the unknown function. Every individual genome or AST can be assigned a fitness number indicating how close the ASTs outputs are to those of the unknown function for each input. The unknown function depends on the problem: it is only known through a Manuscript submitted to ACM Gene Expression Programming for Quantum Computing 3 series of inputs and outputs. When the genetic simulation converges, the “last” population of individual ASTs will all constitute accurate ASTs (that is, computer programs) that output the function our problem required. We could then select the shortest of these ASTs, or one that shows a previously unidentified pattern. The overarching difficulty with the just described approach lies in the combinations and mutations: they do not necessarily create a new valid tree, that is, the resulting AST may not be syntactically correct. A syntactically incorrect tree cannot have a fitness value because it cannot be evaluated, and the approach just described fails. Of course, one could restrict the mutations in complicated ways, but this would have a large and detrimental impact on convergence. That is why non-coding regions are used [8] in the AST, when represented as a string. Figure 1(b) shows how a fixed length string of thirty characters has a coding region of seven characters, and a non-coding region of twenty-three characters. By introducing non-coding regions, all mutations become valid and are allowed, restricting mutations is no longer necessary, and convergence vastly improves. Obviously, the structural organization of genes must be preserved, which is achieved by always maintaining the boundaries between coding and non-coding regions and by not allowing symbols from the function set on the non-coding region. The resulting coding region can be deduced automatically for any string, so that the genotype has been decoupled from the phenotype. With this background in place, we can now list many features of gene expression programming (GEP): multiple outputs by having multiple genes to form a “chromosome,” and multiple chromosomes to form a “cell;” numeric constants can be added and evolved; blocks of “genetic” material can be used in automatically defined functions that aid problem solving; genes can be subjected to large variety of mutations and recombinations. 2.2 Representation of Circuits A0,2 x0 D B1,2 A x1 B C2,3 x2 C D0 x3 ψ0 A02B12C23D0ψ Fig. 2. An example of a 4-bit quantum circuit with gates A, B, C, and D in order to describe the circuit encoding in gene expression programming. Representation of the previous figure’s quantum circuit in GEP. The “primitives” or gates are D0 , C2,3 , B1,2 , and A0,2 . Every primitive takes one input and produces one output. The final output is A0,2 B1,2 C2,3 D0𝜓 0 . To represent the four bit quantum circuit of Fig. 2 in gene expression programming, we consider the Hilbert space H = C2 , with its usual inner product, so that the inputs are in H . Gates are the GEP primitives, and are functions of 𝑁 H into H . One-bit gates act only on one of the 𝑁 bits; there are 𝑁 of them for each type (like 𝑁 Hadamard gates), and the bit they act on is indicated with a subscript. The multiplicity of each type of two-bit gate (like the CNOT gate) is 𝑁 (𝑁 − 1); the multiplicity of each type of three-bit gate (like the CCNOT gate) would be 𝑁 3 − 3𝑁 (𝑁 − 1) − 𝑁 . The gene for Fig. 2 is thus encoded by the string A0,2 B1,2 C2,3 D0𝜓 0 . Manuscript submitted to ACM 4 Gonzalo Alvarez, Ryan Bennink, Stephan Irle, and Jacek Jakowski 2.3 Optimization Procedure QuantumGEP solves two types of problems with two slightly different algorithm variants. The first problem is finding a quantum circuit Cunknown that implements a quantum function given by a set of input-output pairs |𝜓 in 𝑖 ⟩ → |𝜓 𝑖 ⟩ out for 𝑖 = 1, 2, . . . , 𝐷. These input-output pairs constitute training data for the quantum function to be learned. The second problem is finding a quantum circuit C that, when applied to a fixed initial state |𝜓 in ⟩, minimizes the expectation value of some physical observable 𝐻 . An example of this second type of problem is finding the ground state of a many-body Hamiltonian. The gene expression programming procedure to solve either of these two problems is as follows. (For simplicity, the description is written as though for problem 1 there is a single input-output relation |𝜓 in ⟩ → |𝜓 out ⟩ to be satisfied. In the case of multiple input-output pairs, steps 4 and 5 should be modified to compute the average fitness of each circuit over all input-output pairs.) (1) Start with an original population of 𝑀 circuits generated randomly. (2) By gene expression programming mutations and combinations of the existing size 𝑀 population, generate 𝑀 ′ = 2𝑀 new circuits {C(𝜑)}0≤ 𝑗 <𝑀 ′ . These circuits may depend on 𝐾 continuous variables 𝜑 ≡ {𝜑𝑘 }0≤𝑘<𝐾 . To understand why this is so, consider that the rotation gate may depend on the angle of rotation, and, in general, gates may depend on arbitrary parameters that we collectively call 𝜑. (3) For every circuit 𝑗, define the 𝜑−dependent state |𝜓 𝑗 (𝜑)⟩ ≡ C𝑗 (𝜑)|𝜓 in ⟩. (4) For every circuit 𝑗, compute the pre-fitness function 𝑃 𝑗 : |⟨𝜓 out |𝜓 𝑗 (𝜑)⟩| 2 ( for problem 1 𝑃 𝑗 (𝜑) ≡ (1) −⟨𝜓 𝑗 (𝜑)|𝐻 |𝜓 𝑗 (𝜑)⟩ for problem 2 and find the 𝜑 where the maximum occurs; let’s call it 𝜑 max . (5) For every circuit 𝑗, calculate its fitness 𝐹 𝑗 ≡ 𝑃 𝑗 (𝜑 max ). (6) Eliminate the 𝑀 circuits with least fitness and keep the remaing 𝑀 circuits as the source for the next generation. (7) Go to step 3. For the studies described in this paper, 𝑀 = 100. 2.4 Code Structure The free and open source code can be found at https://code.ornl.gov/gonzalo_3/evendim and at https://github.com/ g1257/evendim. QuantumGEP can be regarded as “artificial intelligence (AI)” to distinguish it from the generated ASTs that run inside it: the AI can then be considered also as a virtual machine that runs ASTs in order to calculate their fitness. QuantumGEP can be divided into two coding sections: a computational engine that is independent of fitness function and the set of primitives and inputs (evendim); and a problem-dependent part: a fitness function and a set of primitives and nodes that can be programmed for each particular problem to be solved (our quantum circuit problem solver QuantumGEP). An arithmetic set has been implemented to test compilation on different platforms, and as a fast way to test development ideas. The two types of algorithms discussed in subsection 2.3 create two code paths, distinguished only by C++ templates; both paths are instantiated in the same executable and are chosen at runtime from the input file, by setting RunType to either FunctionFit or GroundState. QuantumGEP reads from the input file all non problem-specific parameters, such as number of generations and population. A parameters object is created, as well as an evolution object, and from both Manuscript submitted to ACM Gene Expression Programming for Quantum Computing 5 an engine object is instantiated with an initial random population; this last object calls its member function evolve() in a loop as many times as the number of generations, with an optional early stop which is useful in cases where the maximum fitness value is known. The evolve() function in the computational engine starts by creating new individuals, in its first call from the initial population, and in subsequent calls from the surviving individuals. It does so by using the following four algorithms in succession: (1) one-point recombination, (2) two-point recombination, (3) mutation, (4) inversion, and (4) swap; all these algorithms were implemented as detailed in [8], and we briefly review them in the following. Recombination involves two parent chromosomes and results in two new individuals. One-point recombination consists of paring the parent chromosomes side by side, choosing a random point at which the parent chromosomes are split up, and exchanging the genetic content after the recombination point between the two chromosomes. Two-point recombination pairs the chromosomes side by side as before, chooses two random points, and exchanges the genetic material between these two points, creating two new individuals. A mutation changes one character of the string representation of the chromosome; in the head any character can change to any other, so any function can be changed to any other without regards to the number of arguments. In the tail, terminal or leafs are changed only into terminal or leafs so that the head and tail structure of the chromosome is preserved by the mutation. Inversion involves inverting the characters in the head of the chromosome, and does not affect the tail. evendim inverts the complete head even though subsets of the head could be inverted instead. Finally, a swap exchanges two characters in the string representation of a chromosome such that the head and tail structure is preserved. After the new population has been created, which also includes the surviving individuals from the previous generation, an optional canonicalization procedure is applied. For quantum circuits, the canonicalization orders the gates by the bit or bits they acts on; there is also here an opportunity for symbolic simplifications: for example, the Pauli matrix gate 𝜎 𝑧 if applied on the same bit twice yields the identity. More complicated simplifications could be added here as well, based on commutation rules among operators or gates. Finally, QuantumGEP sorts the individuals by fitness, its definition depending on the problem to be solved as shown in Eq. (1), and discards as many individuals with lowest fitness as needed to obtain the population supplied in the input file. 3 CASE STUDIES 3.1 Graph Maximum Cut Finding the maximal cut of a graph is a well known problem (the MaxCut problem) in quantum computing [5, 9, 11, 13, 18– 20] The maximal cut is a partition of a graph’s vertices into disjoint sets 𝐴 and 𝐵, such that the number of edges between the set 𝐴 and the set 𝐵 is the largest. MaxCut for a graph 𝐺 is equivalent to finding the ground state of the classical Ising model [3] with Hamiltonian Í 𝐻 = 𝑖,𝑗 ∈𝐸 (𝐺) 𝑆𝑖 𝑆 𝑗 , where the sum is over vertices (or “sites”) that form edges in 𝐺, and 𝑆𝑖 ∈ {−1, 1} is the “spin” configuration at vertex 𝑖, such as a radical electron embedded in an atomic lattice. QuantumGEP used with this Hamiltonian excels at solving MaxCut, by which we mean that it excels at finding a quantum circuit that yields the ground state of the graph Ising model from an initial state. We discuss in detail two examples. Figure 3 shows our first example graph; it has 8 vertices. After it converges we see the following circuit: Ry0 :𝜋 Ry1 :𝜋 Ry2 :𝜙 0 P4 P4 with fitness 4.9999(6). We know this circuit is a perfect solution, that is, it is a true ground state because (i) its fitness is 5 within numerical error, and the true ground state of this graph is −5. The maximal cut is found by looking at the basis states with weight greater than a certain threshold 𝜖. For numerical stability we use 𝜖 = 1𝑒 − 4 but Manuscript submitted to ACM 6 Gonzalo Alvarez, Ryan Bennink, Stephan Irle, and Jacek Jakowski the actual value is not crucial. With this criterion, QuantumGEP yields the basis states 3 and 7, that is, two basis states. State 3 is binary 00000011, which means that sites 0 and 1 have negative “spin” and the rest positive “spin.” Similarly, the other solution is 7 (binary 00000111), which means that sites 0, 1, and 2 have negative “spin” and the rest positive “spin.” An initial state needs to be provided, representing a single graph or a superposition of graphs. We have chosen the graph without a cut, that is 00000000, which can also be described as a configuration with all “spins” positive. We have found that the solutions are robust to this initial condition. QuantumGEP allows the specification of the initial step in the input file. Figure 4 shows our second example labeled 4648 in [11]. After it converges we obtain the quantum circuit Ry3 :𝜋 P3 P3 Ry4 :𝜋 Ry5 :𝜋 Ry6 :𝜋 Ry7 :𝜋 with fitness 8.9999(9) and weight only on basis state 248. Following a similar reasoning than before, we conclude that this is a perfect solution circuit, and 248 (binary 1111000), indicates that vertices 3 to 7 have negative signs and form the maximal cut for this graph. 4 1 3 5 0 2 7 6 Fig. 3. Graph 34. A solution is given by 3 or 7. The first number (3) is 00000011 in binary, which means sites 0, 1, have negative signs and form the maximal cut; The other solution found is 7, 00000111 in binary, which corresponds to sites 0, 1, and 2 having negative signs and forming the maximal cut. Vertices 0, 1, and 2 have been shaded, with vertex 2 shaded more to indicate that it appears only in the second solution. In all cases energy equals -5, fitness 5. 3 0 6 1 5 2 7 4 Fig. 4. Graph 35, which is number 4648 in the qaoa paper [11]. A solution is given by 248, which in binary is 11111000, which means sites 3 to 7 have negative signs and form the maximal cut; energy equals -9, fitness 9. 3.2 Condensed Matter Models Í We apply QuantumGEP to the XX model on a linear chain of sites; the model is given by 𝐻 = 𝐽𝑥 𝑖 𝜎𝑖𝑥 𝜎𝑖+1 𝑥 , where 𝐽𝑥 is a real number acting as coupling constant, 𝜎 𝑥 is the Pauli 𝑥 matrix, and the usual notation for tensor product of operators has been used. We aim at finding the quantum circuit that produces the ground state of this model from an initially prepared state. We choose the vector |0000⟩, periodic or open boundary conditions as detailed below, coupling 𝐽𝑥 = 1, and quantum gates Ry and P. All these parameters can be chosen without recompilation from the input file. After a few generations we get various perfect “individuals,” that is, quantum circuits. For example, with periodic boundary conditions we get Ry0 :3𝜋/2 Ry1 :𝜋/2 Ry2 :3𝜋/2 Ry3 :𝜋/2 with fitness 3.9971(6) which yields the exact answer Manuscript submitted to ACM Gene Expression Programming for Quantum Computing 7 2 Best circuit ∆Eb Largest ∆Ep in population 1.5 ·10−2 ∆Eb or ∆Ep 1.5 1 1 0.5 0.5 0 0 50 100 0 0 20 40 60 80 100 Generation Fig. 5. The plus symbols indicate the energy difference between the ground state and that yielded by the best individual at that generation; the open circles indicate the largest energy difference within the population at that generation. The inset shows a magnification of the figure starting at generation eight. for the ground state of four sites, and another state can be obtained by symmetry. With open boundary conditions, we obtain the same vector, but fitness (or minus the energy) equals 3. Figure 5 shows the convergence of QuantumGEP to a perfect circuit: The x-axis is the generation and the y-axis is (i) for the plot indicated by plus symbols, the energy difference Δ𝐸𝑏 between the ground state and that yielded by the best individual of that generation, and (ii) for the plot indicated by open circles, the largest energy difference Δ𝐸𝑝 within the population of that generation. The first generations show large improvements in quantum circuit quality (open circles), and circuit quality continues to improve but with smallest changes in Δ𝐸𝑏 for late generations. In addition, variations in the population of circuits (solid green line) show a large diversity increase for the first generations. Even though later these population variations are much smaller due to evolutionary pressures, the circuits themselves are all different: they just yield either the same energy or an energy very close to the exact one. 3.3 Quantum Chemistry Applications QuantumGEP can find a quantum circuit, which yields the ground state of a quantum chemistry problem, when applied to an initial state. We have considered only basic quantum gates, and have applied QuantumGEP to two molecules: NaH and the benzene molecule. A four bit Hilbert space models the NaH molecule, and the ground state resides in the two electron sector; see [12] for the full quantum computing solution. One basis state accounts for over 99% of the ground state weight: the Hartree Fock solution; three other basis states contribute the rest with a combined weight of less that 1%. QuantumGEP easily yields circuits for the Hartree Fock state. For example, Ry1 : 𝜋 P2 Ry3 : 𝜋 P3 with fitness 160.29(9)), which equals the opposite of the Hartree-Fock energy, whereas the true ground state energy is -160.30137813(6) Hartrees. In our tests, QuantumGEP was unable to produce the other basis states that contribute to the ground state, as we discuss in section 4.1. An eight bit state models the benzene molecule, with the ground state in the four electron sector. Once more, the Hartree-Fock state accounts for most of the weight, this time at over 95%. QuantumGEP easily yields circuits that Manuscript submitted to ACM 8 Gonzalo Alvarez, Ryan Bennink, Stephan Irle, and Jacek Jakowski produce the Hartree Fock solution, such as Ry0 : 𝜋 P0 Ry1 : 𝜋 P2 Ry4 : 𝜋 Ry5 : 𝜋 P7 with fitness 8.89(4) or the opposite of the Hartree-Fock energy. By decreasing the maximum circuit size or Headsize in the input file, even smaller circuits can quickly be found, such as Ry0 : 𝜋 Ry1 : 𝜋 Ry4 : 𝜋 Ry5 : 𝜋. The other contributions to the ground state are more difficult to find, and require enabling conjugate gradient from the input, as well as a rescaling of the energy in the input: 𝐻 ′ = 10(𝐻 − 𝐸𝐻 𝐹 ), where 𝐸𝐻 𝐹 is the Hartree-Fock energy, or values close to it. Obtaining all these small contributions to the ground state will be the focus of future work. 4 CONVERGENCE AND ACCELERATION 4.1 Convergence Failures We have observed a failure of convergence in the case of the NaH system for four bits[12], where the ground state equals −0.99854285211(3)|1010⟩ + · · · , where the ellipsis indicates states with weight to complete the normalization. We surmise that the failure occurs due to the very high weight of a single basis state, and would require for its solution the use of modular gates so that a more sophisticated quantum circuit can be constructed by the artificial intelligence. We have also found a convergence failure in the two-dimensional quantum Heisenberg model on a 3×3 lattice or 9 bits. In this case, we believe the lack of convergence results from the large number of one- and two-bit gates that would be required to express the ground state of the Heisenberg model in this case. We know that the solution can be written in shorter form by using many bit gates in a multi-scale entanglement answer representation [16]. Here again we think the problem would be solved by modular gates. 4.2 Shared Memory Parallelization QuantumGEP presents multiple opportunities for shared memory parallelization, and we have implemented two of them. The need to apply a matrix to a vector leads to the use of BLAS’s [2] ZGEMM call that can be parallelized over multiple cores on done on the GPU; these matrix-vector operations appear in Hamiltonian actions or gate actions. Moreover, the fitness of each quantum circuit of a given generation can be calculated in parallel, and we have implemented this in pure C++ using PsimagLite’s Parallelizer2, which supports either an openMP or a pthreads back-end. 4.3 Source Code and Documentation We have released both the scientific application QuantumGEP and its computational engine evendim at https://code. ornl.gov/gonzalo_3/evendim and at https://github.com/g1257/evendim, with a free and open source license. They are written fully in C++; the documentation, test suite location, dependencies, and other details are given in the supplemental material FIXME: URL will be given by publisher. 5 SUMMARY AND OUTLOOK Quantum circuits have been created manually, and only recently has artificial intelligence been used in this field [17]. An automatic tool can come to the aid of quantum circuit generation, even when manual intuition remains important. This paper has then introduced QuantumGEP, an automatic generator of quantum circuits to produce the ground state of quantum and classical Hamiltonians, based on gene expression programming. QuantumGEP is free and open source, with a computational engine (evendim) agnostic to fitness function, to primitives (or gates) and to leaves (or inputs), so that problem specific fitness and primitives can be written without modifying evendim. Manuscript submitted to ACM Gene Expression Programming for Quantum Computing 9 The richness of GEP [8] allows for constants, automatically defined functions, and other features included in evendim; GEP’s use of “junk DNA” allows for almost unrestricted mutations, improving convergence. QuantumGEP was successfully applied to the generation of circuits for the MaxCut problem, and it was able to yield intuitive answers in simple condensed matter problems. In quantum chemistry it successfully yielded the Hartree-Fock ground eigenstate, but more work remains to obtain more precise quantum circuits for the exact ground state. We think this will entail the use of modular gates, which can be built as combination of basic gates, where automatically defined functions (ADF) and multiple “genes” would improve convergence. Although ADFs and multiple genes are already implemented in evendim, we did not want to expand its application to difficult quantum chemistry problems in this introductory paper, and leave those for future study. ACKNOWLEDGMENTS This work was performed at Oak Ridge National Laboratory, operated by UT-Battelle, LLC under contract DE-AC05- 00OR22725 for the US Department of Energy (DOE). Support for the work came from the DOE Advanced Scientific Computing Research (ASCR) Accelerated Research in Quantum Computing (ARQC) Program under field work proposal ERKJ354. REFERENCES [1] Sanjeev Arora. 2007. Computational complexity. 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