A Novel Quantum Algorithm for Efficient Attractor Search in Gene Regulatory Networks

TL;DR

The paper tackles attractor search in Boolean-network models of Gene Regulatory Networks, an NP-hard problem due to the exponential state space $2^n$. It introduces a Grover-inspired quantum algorithm that iteratively suppresses basins of already found attractors from a uniform superposition, ensuring each run yields a new attractor. The method relies on a five-step amplitude-suppression circuit with quantum counting and a tailored phase shift, yielding exact suppression when $eta = M/N < 3/4$ and a corresponding iteration count $J = \left\lceil \frac{\pi}{2\pi-4\beta}-\frac{1}{2} \right\rceil$, demonstrated on two model BN networks and shown to be robust to NISQ-era noise via quantum-simulator experiments and IBMQ Brisbane noise models. The work suggests practical applicability to larger networks, potential extension to quantum annealers, and provides open-source code for replication and visualization.

Abstract

The description of gene interactions that constantly occur in the cellular environment is an extremely challenging task due to an immense number of degrees of freedom and incomplete knowledge about microscopic details. Hence, a coarse-grained and rather powerful modeling of such dynamics is provided by Boolean Networks (BNs). BNs are dynamical systems composed of Boolean agents and a record of their possible interactions over time. Stable states in these systems are called attractors which are closely related to the cellular expression of biological phenotypes. Identifying the full set of attractors is, therefore, of substantial biological interest. However, for conventional high-performance computing, this problem is plagued by an exponential growth of the dynamic state space. Here, we demonstrate a novel quantum search algorithm inspired by Grover's algorithm to be implemented on quantum computing platforms. The algorithm performs an iterative suppression of states belonging to basins of previously discovered attractors from a uniform superposition, thus increasing the amplitudes of states in basins of yet unknown attractors. This approach guarantees that a new attractor state is measured with each iteration of the algorithm, an optimization not currently achieved by any other algorithm in the literature. Tests of its resistance to noise have also shown promising performance on devices from the current Noise Intermediate Scale Quantum Computing (NISQ) era.

Figures (6)

• Figure 1: : Schematic of a circuit implementing the time evolution dynamics of a given Boolean network. The scheme on the left produces a superposition of all attractors, and the final measurements will lead to finding one of them. On the right, an example of what a time evolution operator can look like for a given set of Boolean rules.
• Figure 2: : Schematic example of the basin suppression algorithm. In this example, drawn for 3 genes for better understanding, we show the suppression of two attractor basins of the network, namely the states $\ket{001}$ and $\ket{111}$. The numbered steps on top refer to the phases defined in Sec.\ref{['sec:algorithm']}, while the estimation of the basin size is omitted for better visualization of the attractors measurement routine. The final measurement (step 2), after the final time-evolution operator, will lead to the measurement of an attractor state not yet uncovered and suppressed.
• Figure 3: : On the left: Schematic representation of the Boolean network generated by 4 interacting genes designed ad hoc as a test case. The attractors are labeled with the order of the experimental measurement in our example. On the right: Count probability of measuring each attractor of the system after each run of the algorithm. The attractors found in the previous run are suppressed in the following runs. For clarity, the histogram highlights "0 counts" results.
• Figure 4: : On the left: Schematic representation of the Boolean network generated by 5 interacting genes described by Giacomantonio et alGiacomantonio2010. Only the two attractors are marked for better readability. On the right: Count probability of measuring each attractor of the system. The three runs ($1^{st}, 2_A^{nd}$ and $2_B^{nd}$) describe respectively the measurement of the first attractor (probabilistic choice of one of the two) and the measurement of the second attractor depending on the result of the previous measurement. For clarity, the histogram highlights "0 counts" results.
• Figure 5: : Subsequent runs of the deletion algorithm over the attractor basins of the 4-gene network introduced above. We prove our algorithm on a simulator that takes into account the noise profile extrapolated by the ibm_brisbane quantum device. In each run, we delete one more attractor basin (assumed to have been measured in the previous run) in the following order 0000 -> 1011 -> 0011 -> 1111. The last one to be measured will be 1101.