Abstract
Lattice protein folding models are a cornerstone of computational biophysics. Although these models are a coarse grained representation, they provide useful insight into the energy landscape of natural proteins. Finding lowenergy threedimensional structures is an intractable problem even in the simplest model, the HydrophobicPolar (HP) model. Description of proteinlike properties are more accurately described by generalized models, such as the one proposed by Miyazawa and Jernigan (MJ), which explicitly take into account the unique interactions among all 20 amino acids. There is theoretical and experimental evidence of the advantage of solving classical optimization problems using quantum annealing over its classical analogue (simulated annealing). In this report, we present a benchmark implementation of quantum annealing for lattice protein folding problems (six different experiments up to 81 superconducting quantum bits). This first implementation of a biophysical problem paves the way towards studying optimization problems in biophysics and statistical mechanics using quantum devices.
Introduction
The search for more efficient optimization algorithms is an important endeavor with prevalence on many disciplines ranging from the social sciences to the physical and natural sciences. Belonging to the latter, the protein folding problem^{1,2,3,4,5,6,7} consists of finding the lowest freeenergy configuration or, equivalently, the native structure of a protein given its aminoacid sequence. Knowing how proteins fold elucidate their threedimensional structurefunction relationship which is crucial to the understanding of enzymes and for the treatment of misfoldedprotein diseases such as Alzheimer's, Huntington's and Parkinson's disease. Due to the high computational cost of modeling proteins in atomistic detail^{8,9}, coarsegrained descriptions of the protein folding problem, such as those found in lattice models, provide valuable insight about the folding mechanisms^{2,4,5,6,10}.
Harnessing quantummechanical effects to speed up the solving of classical optimization problems is at the heart of quantum annealing algorithms (QA)^{11,12,13,14,15}. There is theoretical^{11,12,16,17,18} and experimental^{19} evidence of the advantage of solving classical optimization problems using QA^{11,12,13,14} over its classical analogue (simulated annealing^{20}). In QA, quantum mechanical tunneling allows for more efficient exploration of difficult potential energy landscapes such as that of classical spinglass problems. In our implementation of lattice folding, quantum fluctuations (tunneling) occurs between states representing different model protein conformations or folds.
The theoretical challenge is to efficiently map the hard computational problem of interest (e.g., lattice folding) to a classical spinglass Hamiltonian: such mapping requiring a polynomial number of quantum bits (qubits) with the size of the problem (protein length) is described elsewhere^{21}. Here we present a new mapping which, due to its exponential scaling with problem size, is not intended for large instances. The proposed mapping employs very few qubits for small problem instances, making it ideal for this first experimental demonstration and implementation on current quantum devices^{22}. A combination of the existing polynomial mapping^{21} and more advanced quantum devices would allow for the simulation of much larger instances of lattice folding and other related optimization problems.
Solving arbitrary problem instances requires a programmable quantum device to implement the corresponding classical Hamiltonian. We employ quantum annealing on the programmable device to obtain lowenergy conformations of the protein model. We emphasize that nothing quantum mechanical is implied about the protein or its folding process; rather quantum fluctuations are a tool we use to solve the optimization problem.
The QA protocol performed here is also known as adiabatic quantum computation (AQC)^{17,23}. Of all the quantumcomputational models, AQC is perhaps the most naturally suited for studying and solving optimization problems^{17,24}. For the experiments presented here, the small finite temperature of the superconducting device is enough to make the process less coherent than the original formulation of AQC, where the theoretical limit of zero temperature and quasiadiabaticity are usually assumed^{17,23}. As we show in the discussion, numerical simulations including these unavoidable environmental effects accurately reproduce our experimental results.
Experimental implementations of QA or AQC are limited either by the number of qubits available in stateoftheart quantum devices or by the programmability required to fulfill the problem specification. For example, the first realization of AQC was performed on a threequbit NMR quantum device^{25} and newer NMR implementations involve four qubit experiments^{26}. Other experimental realizations of spin systems have been based on measuring bulk magnetization properties of the systems in which there is no control over the individual spins and the couplings among them^{19,27,28}. Quantum architectures using superconducting qubits^{29,30,31,32,33,34,35,36} offer promising device scalability while maintaining the ability to control individual qubits and the strength of their interaction couplings. During the preparation of this manuscript, an 84qubit experimental determination of Ramsey numbers with quantum annealing was performed^{37}, underscoring the programmable capabilities of the device for problems with over 80 qubits. In this letter, we present a quantum annealing experimental implementation of lattice protein models with general (MiyazawaJernigan^{38}) interactions among the amino acids. Even though the cases presented here still can be solved on a classical computer by exact enumeration (the sixaminoacid problem has only 40 possible configurations), it is remarkable that the device anneals to the ground state of a search space of 2^{81} possible computational outcomes. This study provides a proofofprinciple that optimization of biophysical problems such as protein folding can be studied using quantum mechanical devices.
Results
The quantum hardware employed consists of 16 units of a recently characterized eightqubit unit cell^{22,39}. Postfabrication characterization determined that only 115 qubits out of the 128 qubit array can be reliably used for computation (see Fig. 1). The array of coupled superconducting flux qubits is, effectively, an artificial Ising spin system with programmable spinspin couplings and transverse magnetic fields. It is designed to solve instances of the following (NPhard^{40}) classical optimization problem: Given a set of local longitudinal fields {h_{i}} and an interaction matrix {J_{ij}}, find the assignment , that minimizes the objective function E(s), where,
h_{i} ≤ 1, J_{ij} ≤ 1 and s_{i} ∈ {+1, 1}.
Finding the optimal s* is equivalent to finding the ground state of the corresponding Ising classical Hamiltonian,
where are Pauli matrices acting on the ith spin.
Experimentally, the timedependent quantum Hamiltonian implemented in the superconductingqubit array is given by,
with responsible for quantum tunneling among the localized classical states, which correspond to the eigenstates of H_{p} (the computational basis). The timedependent functions A(τ) and B(τ) are such that A(0) ≫ B(0) and A(1) ≪ B(1); in Fig. 2(b), we plot these functions as implemented in the experiment. t_{run} denotes the time elapsed between the preparation of the initial state and the measurement.
QA exploits the adiabatic theorem of quantum mechanics, which states that a quantum system initialized in the ground state of a timedependent Hamiltonian remains in the instantaneous ground state, as long as it is driven sufficiently slowly. Since the ground state of H_{p} encodes the solution to the optimization problem, the idea behind QA is to adiabatically prepare this ground state by initializing the quantum system in the easytoprepare ground state of H_{b}, which corresponds to a superposition of all 2^{N} states of the computational basis. The system is driven slowly to the problem Hamiltonian, H(τ = 1) ≈ H_{p}. Deviations from the groundstate are expected due to deviations from adiabaticity, as well as thermal noise and imperfections in the implementation of the Hamiltonian.
The first challenge of the experimental implementation is to map the computational problem of interest into the binary quadratic expression (Eq. 2), which we outline next. In lattice folding, the sequence of amino acids defining the protein is viewed as a sequence of beads (amino acids) connected by strings (peptide bonds). This bead chain occupies points on a two or threedimensional lattice. A valid configuration is a selfavoiding walk on the lattice and its energy is calculated from the sum of interaction energies between nearest nonbonded neighbors on the lattice. By the thermodynamic hypothesis of protein folding^{41}, the global minimum of the freeenergy function is conjectured to be the native functional conformation of the protein.
Finding lowenergy threedimensional structures is an intractable problem^{42,43,44}. The hydrophobicpolar (HP) model is one of the simplest possible models for lattice folding^{45}. In this model, the amino acids are classified into two groups, hydrophobic (H) and polar (P). Even in this simplest model, exhaustive search of all possible global minima is limited to sequences in the tens of amino acids^{46}. To describe real protein energy landscapes a more elaborate description needs to be considered, such as the MijazawaJernigan (MJ) model^{38} which assigns the interaction energies for pairwise interactions among all twenty amino acids. The formulation we used is general enough to take into account arbitrary interaction matrices for lattice models in two and three dimensions. In particular, we solved a MJ model in 2D, the sixaminoacid sequence of ProlineSerineValineLysineMethionineAlanine (PSVKMA in the oneletter aminoacid sequence notation). We solved the problem under two different experimental schemes (see Schemes 2 and 3 in Fig. 3), each requiring a different number of resources. Solving the problem in one proposed experimental realization (Scheme 1) requires more resources than the number of qubits available (115 qubits) in the device. Scheme 2 and 3 are examples of the divideandconquer strategy, in which one partitions the problem in smaller instances and combines the independent set of results, thereby obtaining the same solution for the intractable problem. In the Supplementary Information section, we complement these four MJ related experiments with two small tetrapeptide instances (effectively HP model instances) for a total of six different problem Hamiltonians. We used the largest of these two instances (an 8 qubit experiment) for direct theoretical simulation of the annealing dynamics of the device. The results from our experiment and the theoretical model, which does not use any adjustable parameters (all are extracted experimentally from the device), are in excellent agreement (see Supplementary Fig. S2 online).
To represent each of the possible Naminoacid configurations (folds) in the lattice, we encode the direction of each successive bond between amino acids; thus, for every Nbead sequence we need to specify N  1 turns corresponding to the number of bonds. For the case of a two dimensional lattice, a bond can take any of four possible directions; therefore, two bits per bond are required to uniquely determine a direction. More specifically, if a bond points upwards, we write “11”. If it points downwards, leftwards or rightwards, we write “00”, “10”, or “01” respectively. Fixing the direction of the first bond reduces the description of any Nbead fold to binary variables, without loss of generality. As shown in Fig. 2(a), in the absence of external constraints other than those imposed by the primary aminoacid sequence (see Supplementary Information for an example with external constraints), we can fix the third binary variable to “0”, forcing the third amino acid to go either straight or downward and reducing the number of needed variables to . This constraint reduces the solution space by removing conformations which are degenerate due to rotational symmetry. Thus, a particular fold is uniquely defined by,
An example of this encoding for a sixaminoacid sequence is represented in Fig. 2(a).
Using this mapping to translate between the aminoacid chain in the lattice and the 2(N  1) string of bits, we constructed the energy function E(q) in which q denotes the remaining 2N5 binary variables. Additionally, we penalized folds which exhibit two amino acids on top of each other, to favor selfavoiding walk configurations. The energy penalty chosen for each problem was sufficient to push the energy of invalid folds outside of the energy range of valid configurations (those with E ≤ 0). Finally, we took into account the interaction energy among the different amino acids. A detailed construction of our energy function for the general case of N amino acids with arbitrary interactions is given elsewhere.
The experiment consists of the following steps: a) construction of the energy function to be minimized in terms of the turn encoding; b) reduction of the energy expression to a twobody Hamiltonian; and finally, c) embedding in the device. These last two steps need additional resources as explained below. We will focus on the simplest example (Experiment 3, Fig. 3) to show the procedure in detail. The embeddings for the other five experiments are provided in the Supplementary material. The energy function for Experiment 3, containing the contributions due to onsite penalties for overlapping amino acids and pairwise interactions between amino acids is,
where q_{1}0 (q_{2}q_{3}) encodes the orientation of the fourth (fifth) bond (see Fig. 3). From Eq. 5 one can verify by substitution that the eight possible threebitvariable assignments provide the desired energy landscape: the six conformations with E ≤ 0 shown in blue in Fig. 3.
Eq. 5 describes the energy landscape of configurations but it is not quite ready for the device. Experimentally, we can specify up to twobody spin interactions, and therefore, we need to convert this cubic energy function (Eq. 5) into a quadratic form resembling Eq. 1 (see Supplementary Information for details). The resulting expression is
where the original binary variables and spin operators are related by . Experimental measurements of yield s_{i} = +1 (s_{i} = −1) corresponding to q_{i} = 0 (q_{i} = 1). Since q_{i} = (1 − s_{i})/2, measurement of s_{1}, s_{2} and s_{3} allows us to reconstruct the bit string q_{1}0q_{2}q_{3} which encodes the desired fold.
One ancilla variable was added during the transformation of the threevariable cubic Hamiltonian into this quadratic fourvariable expression. The meaning of the original variables s_{1}, s_{2} and s_{3} remains the same, allowing for the reconstruction of the folds. The energy of this fourvariable expression will not change as long as the measurements of through result in values for q_{1}q_{2}q_{3}q_{4} satisfying q_{4} = q_{2}q_{3}. This transformation ensures an energy penalty whenever this condition is violated.
The architecture of the chip lacks sufficient connectivity between the superconducting rings for a onetoone assignment of variables to qubits (see Fig. 4). To satisfy the connectivity requirements of the fourvariable energy function, the couplings of one of the most connected variables, q_{4}, were fulfilled by duplicating this variable inside the device such that q_{4} → q_{4} and q_{4′}. In the form of Eq. 2 the final expression representing the energy function of Experiment 3 is given by,
This expression satisfies all requirements for the problem Hamiltonian (Eq. 3), the completion of which allows for the measurement of the energetic minimum conformation of this small peptide instance. The embedding of Eq. 7 into the hardware is shown in Fig. 4, where we label the five qubits used, q_{1}, q_{2}, q_{3}, q_{4} and q_{4′}. Since we want the two qubits representing q_{4} to end up with the same value, we apply the maximum ferromagnetic coupling (J = −1) between them, which adds a penalty whenever this equality is violated (last term in Eq. 7). These maximum couplings are indicated in Fig. 4 by heavy lines. The thinner lines show the remaining couplings used to realize the quadratic terms in Eq. 7, color coded according to the sign of the interaction and its thickness representing their strength. Note that every quadratic term in Eq. 7 has a corresponding coupler. Hereafter, we will denote the outcome of the fivequbit measurements as q_{exp3} = 010010q_{1}0q_{2}q_{3}q_{4}q_{4′}, with q_{i} = 0 (q_{i} = 1) whenever s_{i} = 1 (s_{i} = −1). Notice that only the bits preceding the divider character  contain physical information. These are the ones shown under each of the protein fold drawings associated with Experiment 3 (see Fig. 3).
Similar embedding procedures to the one previously described were used for the larger experiments. For example, in Experiment 1, only 5 qubits define solutions of the computational problem. We needed 5 auxiliary qubits to transform the expression with 5body interactions into an expression with only 2body interactions. Embedding of this final expression required an additional of 18 qubits to satisfy the hardware connectivity requirements, for a total of 28 qubits. Table S1 in the Supplementary material summarizes the number of qubits required in each step through to the final experimental realizations.
Discussion
Even though the quantum device follows a quantum annealing protocol, the odds of measuring the ground state are not necessarily high. For example, in the 81 qubit experiment, only 13 out of 10,000 measurements yielded the desired solution. We attribute these lowpercentages to the analog nature of the device and to precision limitations in the real values of the local fields and couplings among the qubits in the experimental setup. When compared to other problem implementations, physical problems such as lattice folding lack the structure of the Ramsey number problem^{37}. In the lattice folding problem implemented here, the parameters defining the problem instances are arbitrary and do not fall into certain integral distinct values as in the case of the Ramsey number experiment, making precision issues more pronounced in our implementation.
To gain insights into the dynamics and evolution of the quantum system, we numerically simulated the superconducting array with a BlochRedfield model of the 8qubit experiment (see Supplementary material) which takes into account thermal fluctuations in the states due to the finite temperature (20mK) of the quantum device. For this 8qubit experiment, the simulation predicted a ground state probability of 80.7 %, in excellent agreement with the experimentally observed value (80.3%). It is important to note that no adjustable parameters were used in our simulations to fit the data and all the parameters correspond to values measured directly from the quantum device. More details about the numerical simulations can be found in the Supplementary Information.
As seen in Fig. 2(c), the temperature of the device is comparable with the minimum gap of the eightqubit Hamiltonian. Therefore, we expect stronger excitation/relaxation near the gap closing, τ ≈ 0.6, due to exchange of energy with the environment, when compared to the other regimes of the annealing schedule where the gap is much larger than k_{B}T. In the absence of environment (a fully coherent process), our simulations indicate that that the success probability would be 100%, within numerical error. Fig. 2(d) shows that for the simulations at 20 mK, the probability in the ground state goes down to ~ 55%, but the same fluctuations make the system relax back to the ground state, yielding tan 80.27% success probability. This is due to the advantageous natural tendency of the system to approach a thermal equilibrium which favors the ground state after crossing the minimum energy gap. As previously discussed in similar numerical simulations of quantum annealing algorithms^{47}, strong coupling to the bath and nonMarkovianity would require going beyond the BlochRedfield model^{48}, but the agreement between experimental and simulated results support the validity of the quantum mechanical model used to describe the device. Previously reported temperature dependence predictions for the tunneling rate on the same qubits^{22} and excellent agreement with the same level of theory used here reinforce the validity of our simulations for this 8qubit instances.
We present the first quantummechanical implementation of lattice protein models using a programmable quantum device. We were able to encode and to solve the global minima solution for a small tetrapeptide and hexapeptide chain under several experimental schemes involving 5 and 8 qubits for the fouraminoacid sequence (HydrophobicPolar model) and 5, 27, 28 and 81 qubits experiments for the sixaminoacid sequence under the MiyazawaJernigan model for general pairwise interactions. For the experiment with 8 qubits, we simulated the dynamics of the quantum device with a Redfield equation with no adjustable parameters, obtaining excellent agreement with experiment. Since the quantum annealing algorithm not only finds the ground state but also the lowlying excited states, it provides information about the relevant minimum energy compact structures of protein sequences^{49} and it is useful to evaluate designability and stability such as that found in natural protein sequences, where the global minimum of free energy is well separated in energy from other misfolded states^{41}. The approach employed here can be extended to treat other problems in biophysics and statistical mechanics such as molecular recognition, protein design and sequence alignment^{50}.
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Acknowledgements
This work was supported by NSF CCI center, “Quantum Information for Quantum Chemistry(QIQC)”, Award number CHE1037992. The authors thank Sergio Boixo, Mohammad Amin and Ryan Babbush for helpful discussions and revisions of the manuscript.
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A.PO and A.AG developed the theoretical constructs and designed the experiments. M.DB and G.R ran the experiments. N.D ran the BlochRedfield numerical simulations. A.PO, N.D and A.AG wrote the manuscript and the Supplementary Information. All the authors revised the manuscript.
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PerdomoOrtiz, A., Dickson, N., DrewBrook, M. et al. Finding lowenergy conformations of lattice protein models by quantum annealing. Sci Rep 2, 571 (2012). https://doi.org/10.1038/srep00571
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