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Finding low-energy conformations of lattice protein models by quantum annealing
Alejandro Perdomo-Ortiz, Neil Dickson, Marshall Drew-Brook, Geordie Rose, Alán Aspuru-Guzik
TL;DR
The paper addresses the challenge of efficiently mapping a hard optimization problem and presents the first quantum-mechanical implementation of lattice protein models with general Miyazawa-Jernigan interactions. Experiments on small peptide chains included a predicted ground-state probability of 80.7%, in excellent agreement with experiment.
Problem
Efficiently mapping hard computational optimization problems remains a theoretical challenge motivating improved algorithms.
Method
The paper implements lattice protein models with general Miyazawa-Jernigan interactions on a quantum device, studying small tetrapeptide and hexapeptide chains under several experimental schemes.
Results
80.7% predicted ground-state probability was in excellent agreement with the experiment.
Takeaways & Limitations
The implementation supports studying optimization problems in biophysics and statistical mechanics, including molecular recognition and protein design.
Takeaways & Limitations
Ground-state probabilities can be low because of the device’s analog nature and precision limitations.
Abstract
from arXiv · showhide
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 low-energy three-dimensional structures is an intractable problem even in the simplest model, the Hydrophobic-Polar (HP) model. Exhaustive search of all possible global minima is limited to sequences in the tens of amino acids. Description of protein-like 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 a biophysical problem (six different experiments up to 81 superconducting quantum bits). Although the cases presented here can be solved in a classical computer, we present the first implementation of lattice protein folding on a quantum device under the Miyazawa-Jernigan model. This paves the way towards studying optimization problems in biophysics and statistical mechanics using quantum devices.
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The paper maps lattice protein folding to programmable quantum annealing and demonstrates low-energy conformation searches for small protein models. Experiments and parameter-free simulations agree, while current mappings remain limited in scale.
- Scope: The mapping is not intended for large instances because its resource requirements scale exponentially with problem size.The authors suggest combining polynomial mapping approaches with more advanced devices to simulate larger instances.
- Approach: Quantum annealing is used as a tool for solving lattice-folding optimization, not as a quantum model of protein folding.The implementation uses quantum fluctuations to explore configurations of a classical protein model.
- Approach: The implementation supports arbitrary interaction matrices, including general Miyazawa–Jernigan interactions among amino acids.This extends beyond the simplest hydrophobic-polar lattice model.
- Results: 80.7% predicted ground-state probability closely matched the experimentally observed 80.3% for the 8-qubit experiment.The agreement was obtained without adjustable parameters in the simulation.
SUMMARY
The paper presents a quantum-annealing implementation of generalized lattice protein folding using Miyazawa–Jernigan interactions. It describes the energy-function construction, experimental realization, and supporting quantum simulations.
- The study implements lattice protein models with Miyazawa–Jernigan pairwise interactions on a quantum annealing device.
- The paper presents the first experimental implementation of lattice protein folding on a quantum device under the generalized interaction model.
- The energy landscape is reformulated into a succinct expression suitable for implementation in the quantum device.
- Six experimental realizations of the generalized lattice-folding model are described using Miyazawa–Jernigan interactions.
- The paper describes the quantum device and numerical simulations used to support the experimental results.
I. TRANSFORMATION OF THE ENERGY FUNCTION OF THE LATTICE-FOLDING MODEL INTO THE EXPERIMENTALLY
The lattice-model energy is decomposed into onsite, pairwise, and external-potential contributions before being prepared for quantum-device implementation. For sequences in vacuo, only onsite and pairwise terms are required, and the section focuses on two-dimensional energy functions.
- The lattice-model energy is expressed as the sum of onsite, pairwise-interaction, and external-potential contributions.
- The onsite term penalizes configurations in which any two amino acids overlap.
- The pairwise term accounts for nearest-neighbor interactions between non-bonded amino acids.
- The external-potential term represents potentials beyond amino-acid interaction energies defining the protein.
- For amino-acid sequences in vacuo, only the onsite and pairwise terms are needed.
- The section focuses on two-dimensional energy functions, while the general construction applies to arbitrary sequence lengths and interactions in two and three dimensions.
A. Case of the six-amino acid sequence PSVKMA (Experiments 1-4)
The PSVKMA six-amino-acid sequence is represented on a two-dimensional lattice through several reduced schemes, yielding four experiments with experimentally measured state probabilities.
- The complete two-dimensional fold description uses a seven-bit energy function for the six-amino-acid sequence.
- PSVKMA denotes the six-amino-acid sequence Proline-Serine-Valine-Lysine-Methionine-Alanine.
- Experiments 1 through 4 independently solve PSVKMA using three divide-and-conquer schemes.
- 32.70%, 59.88%, 8.00%, and 95.97% are the measured percentages for states with energy greater than zero in Experiments 1 through 4, respectively.
- The four experiments are obtained by fixing selected variables in the complete energy function and relabeling the remaining variables.
B. Case of the four-amino acid sequence HPPH (Experiment 5)
Experiment 5 models the four-amino-acid HPPH sequence within the HP lattice-protein model and specifies its conformational energy landscape with three binary variables.
- HPPH is the four-amino-acid sequence used for Experiment 5 in the HP model.
- For N = 4, any fold is specified by the bit string qexp5 = 010q1q2q3.
- Experiment 5 uses a three-bit energy function to describe the HPPH conformational energy landscape.
(Experiment 6)
Experiment 6 extends the HPPH model with an external chaperone-like potential, requiring a higher-order energy reduction before quantum-device implementation.
- The chaperone-like environment adds an external potential to the intrinsic HPPH chain interactions.
- Adding the external potential removes the upper/lower-half-plane symmetry that simplified the in-vacuo search space.
- Overlaps with the chaperone raise energy by four units, while amino-acid chain crossings raise energy by two units.
- High-order terms are reduced to a 2-body Ising-like Hamiltonian because many-body interactions are not experimentally feasible on the current device.
- Two ancilla variables q5 and q6 replace q1q2 and q3q4, transforming the quartic term into a quadratic interaction.
- The ancilla penalty is zero when the substituted variables satisfy the corresponding AND conditions and positive otherwise.
B. Embedding into the quantum hardware
The reduced Ising expressions are embedded into the quantum hardware by adding auxiliary qubits and couplings that satisfy device connectivity constraints.
- Eq. S16 violates the chip-connectivity requirements for the primal graph.
- The connectivity limitation is addressed by adding two new qubits and redistributing connections among original and primed qubits.
- Strong ferromagnetic couplings penalize disagreements between replicated qubits, preserving the intended assignments in the energy region E ≤0.
- The embedded representations are shown for Experiment 5 as a vacuo HPPH problem and Experiment 6 as an eight-qubit chaperone-like problem.
- The final experiment column reports the number of qubits in the experimentally implemented energy-function expression.
A. The processor chip
The processor is a 128-qubit superconducting array built from coupled eight-qubit unit cells, with tunable couplers and extensive magnetic-field compensation. Experiments used multiple chips sharing the same architecture.
- Experimental hardware: The chips were fabricated using a four-layer niobium superconducting integrated-circuit process and operated at 20 mK.The fabrication included Nb/AlOx/Nb trilayers, TiPt resistors, and planarized SiO2 dielectric layers.
- Chip architecture: The sample processor contains a coupled array of 128 superconducting qubits arranged as 16 eight-qubit unit cells.The array architecture and unit-cell connectivity support the embedded Ising problems used in the experiments.
- Embedding: Embedding adds qubits and couplings so Ising variables lacking direct hardware connections can be represented by strongly ferromagnetically coupled qubit groups.The resulting spin-glass Hamiltonians included ten qubits after reduction, with additional qubits used for embedding.
- Experimental hardware: Six problem instances were distributed across three chips, whose common architecture was accompanied by different calibration parameters.Experiments 1, 2, and 4 used one chip; Experiments 3 and 5 another; Experiment 6 used a third configuration.
- Chip architecture: Each unit cell provides complete bipartite K4,4 connectivity with continuously tunable ferromagnetic or antiferromagnetic couplers.Selective coupler settings can also implement ferromagnetic Ising chains.
C. Experimental method
The experimental method calibrates and homogenizes the qubits, programs embedded Ising Hamiltonians, performs annealing, and reads spin states repeatedly to obtain outcome statistics.
- Procedure: The procedure comprises calibration, homogenization, Hamiltonian programming, repeated annealing, and spin-state readout.The experiment outline identifies calibration and homogenization as initial steps, followed by reading the spin states after annealing.
- Calibration: Calibration extracts intrinsic qubit parameters, including junction critical current and inductance, from circulating-current and tunnelling measurements.Effective couplings are determined by measuring flux changes between opposite states of coupled qubits.
- Sampling: Experiment 6 initially comprised 32,768 repetitions, but the first 512 after each programming step were discarded because programming heated the chip.The retained dataset was 8 × 3,584 = 28,672 repetitions.
- Annealing and readout: Annealing raised the single-qubit tunnelling barrier by changing Φ2x linearly from 0.592 Φ0 to 0.652 Φ0 over 148 µs.After annealing, spin states were measured with hysteretic dc-SQUID readout.
D. Thermometry
Device thermometry compares independent temperature estimates with the mixing-chamber thermometer, while simulations model the calibrated rf-SQUID dynamics and environmental coupling.
- Simulation: The simulation uses calibrated circuit parameters from circulating-current, tunnel-splitting, and MRT peak-spacing measurements.The parameters used in simulation are summarized in Table III.
- Thermometry: Effective device temperature Tth is extracted by fitting equilibrium spin-up probabilities versus Φ1x at fixed Φ2x.The fit uses the circulating current Ip measured at that barrier height.
- Thermometry: The Tth and TMRT estimates generally agree with the mixing-chamber thermometer to within 3 mK over the experimental temperature range.The comparison includes thermometry based on equilibrium probabilities and Macroscopic Resonant Tunnelling transition-rate widths.
- Simulation validation: The model includes coupling to environmental noise, and its predicted evolution uses the same Hamiltonian, environmental coupling, and noise spectral-density type as the experiment.This setup is used to compare simulated and measured low-energy-conformation probabilities.
- Circuit model: The rf-SQUID model retains two flux degrees of freedom and includes mutual-inductance coupling between qubits.The realistic calculations do not neglect the second loop flux Φ2 because the relevant loop inductance is not sufficiently small.
- Quantum dynamics: The quantum dynamics are computed by diagonalizing each single-rf-SQUID Hamiltonian and retaining a low-energy localized-state subspace.The resulting effective Hamiltonian contains tunnelling terms between opposite wells but no matrix elements between states in the same well.