Experimental implementation of a discrete-time quantum walk on biological networks

TL;DR

An algorithm is introduced that leverages symmetry-sector encoding and trades circuit depth for qubits, while integrating symmetry-respecting postselection as an effective noise-mitigation strategy that enables us to execute practical quantum-walk circuits for biological networks on actual quantum hardware.

Abstract

Quantum walks provide a versatile framework for probing the structural and dynamical properties of complex systems ranging from biological networks to synthetic materials. However, their realization on current noisy pre-fault-tolerant quantum computers is fundamentally limited by decoherence. Conventional dense encodings of graph structures require prohibitively deep circuits, making them incompatible with existing hardware. Here we introduce an algorithm that leverages symmetry-sector encoding and trades circuit depth for qubits, while integrating symmetry-respecting postselection as an effective noise-mitigation strategy. This combination enables us to execute practical quantum-walk circuits for biological networks on actual quantum hardware. We benchmark the proposed methodology against known state-of-the-art circuit architectures, highlighting significant reduction of circuit depth in our approach at the cost of moderate qubit overhead. Utilizing 40 qubits, we implement quantum walks on complex graphs containing up to 17 nodes and 20 edges -- the largest experiment on superconducting hardware to date, with the Hellinger fidelity exceeding 87% throughout 7 steps. We present a case study that illustrates how experimentally obtained quantum-walk dynamics on a protein-protein-interaction network can be applied to prioritizing disease-associated genes. We discuss the framework scalability in the pre-fault-tolerant era and its potential for studying larger biological networks.

Figures (6)

• Figure 1: Implementation framework.(a) Example 4-node graph. Nodes 1, 2, 3, and 4 have degree 1, 3, 2, and 2, respectively. (b) Encoding DTQW states $\ket{i \rightarrow j}$ into qubit states. The length of qubit register is twice the number of graph edges. (c) Single-step in DTQW circuit consisting of the shift and coin operators. The shift operator $S$ represents a collection of partial-swap gates associated with opposite directions within the same edge, whereas the coin operator $C$ is composed of the smaller Grover coins $C_i$ associated with the graph nodes $i$ of degree $k_i$. Qubits carry information about excitations (walker) in the corresponding graph edges.
• Figure 2: Quantum walk on an 11-node, 12-edge graph (24-qubit encoding).(a) Subgraph extracted from the BioPlex 3.0 PPI network. Nodes 1--11 correspond to genes PON2, VAMP5, MTOR, LYPD3, FLVCR1, HLA-C, HLA-DQA1, HLA-G, TMEM214, FAM234B, and ADPGK, respectively. Quantum walker is initialized at the highlighted node. (b) 24-qubit circuit layout for the experiment performed on ibm_kingston with heavy-hex connectivity. (c) Boxplot of absolute errors between experimental and simulated probabilities across nodes per step. (d) Probability distribution at Step 1. Brown vertical arrows show the change between paired probabilities introduced by postselection for each node. (e) Fidelity with and without postselection. (f) Postselection ratio across steps.
• Figure 3: Quantum walk on a 15-node, 18-edge graph (36-qubit encoding).(a) Subgraph extracted from the BioPlex 3.0 protein–protein interaction network. Nodes 1--15 correspond to genes IGF2BP3, APOBEC3D, APOBEC3F, SF3B1, TRUB2, ABT1, ZFR, RBMS2, SNRPA1, SNRPB2, LIN28A, COIL, PHF5A, YTHDC1, SART3, respectively. Quantum walker is initialized at the highlighted node. (b) 36-qubit circuit layout for the experiment performed on ibm_pittsburgh with heavy-hex connectivity. (c) Boxplot of absolute errors between experimental and simulated probabilities across nodes per step. (d) Probability distribution at Step 1. Brown vertical arrows show the change between paired probabilities introduced by postselection for each node. (e) Fidelity with and without postselection. (f) Postselection ratio across steps.
• Figure 4: Quantum walk on a 17-node, 20-edge graph (40-qubit encoding).(a) Subgraph extracted from the BioPlex 3.0 PPI network. Nodes 1--17 correspond to genes AHCYL1, ATF7, AGAP3, ATF2, EEF1D, GPC4, AHCYL2, FOSL2, GPC1, HADHA, NFKBIL1, BATF3, CAMKV, FOSL1, KBTBD7, RTL6, CREB5, respectively. Quantum walker is initialized at the highlighted node. (b) 40-qubit circuit layout for the experiment performed on ibm_kingston with heavy-hex connectivity. (c) Boxplot of absolute errors between experimental and simulated probabilities across nodes per step. (d) Probability distribution at Step 1. Brown vertical arrows show the change between paired probabilities introduced by postselection for each node. (e) Fidelity with and without postselection. (f) Postselection ratio across steps.
• Figure 5: Disease-gene prioritization from experimentally measured quantum-walk dynamics.(a) Heatmap shows postselected transition probabilities in DTQWs on a graph depicted in Fig. \ref{['fig:probs_4']}(a). Arrows indicate how strongly quantum interference modifies the probability at node $i$ relative to the classical random walk, and black squares mark the maximum quantum interference index (score). (b) Corresponding biological network with node colors reflecting their scores and the four highest-score genes highlighted in red.