## Introduction

One of the most profound observations regarding the natural world is that, despite the simple set of physical laws that underpin it, the universe displays a plethora of complex, emergent phenomena, encountered in fields as diverse as biology, sociology, and physics1,2,3. Examples of classical systems where complexity arises as a result of many interacting degrees of freedom are ecosystems, the human brain, and power grids4. Certain classical cellular automata (CA) show how complexity can arise from simple rules without the controlling hand of a designer5. CA possess the ability to generate oscillatory, self-replicating structures and in some instances are themselves Turing complete6,7,8,9,10.

It is known, however, that the laws constituting our best model of the universe are quantum mechanical rather than classical11. Therefore, in order to simulate the emergence of complexity more fundamentally, one ought to investigate computational models that are predicated upon quantum mechanics. Goldilocks quantum cellular automata (QCA)12, are a class of computational models that exhibit emergent complexity despite being constructed from repeated blocks of simple local unitary operators13. They involve trade-offs in the local neighborhood such as are known to be sources of complexity in classical systems and essential to self-organized criticality14. Some Goldilocks QCA have been shown to generate mutual information networks that exhibit signatures of complexity, such as large network clustering, short average path length, and broad node-strength distribution, typically only observed in classical, small-world networks like social or biological networks12. In addition, QCA have been proposed for other applications such as lattice discretization in the simulation of strongly-correlated matter, quantum field, and gravitational theories15,16,17. However, the categorical limitation on the ability of classical computers to simulate the time evolution of large quantum systems is a bottleneck for the discovery and exploration of QCA more generally, hampering the theoretical illumination of the class of systems as a whole18. Meanwhile, the last few years have seen the creation of sizeable digital quantum processors that are already demonstrating their value as tools of scientific discovery19,20,21,22,23,24,25. Due to their universality, such processors are ideal platforms on which to elucidate the physics and complexity characteristics of QCA.

Herein, we simulate a particular one-dimensional QCA on a Sycamore-class superconducting processor depicted schematically in Fig. 1a–d. Through the calculation of population dynamics and a complex network characterization of the two-body mutual information matrix, we establish that such QCA form small-world mutual information networks and thereby exhibit emergent physical complexity. Our results contribute to enabling the widespread use of near-term quantum processors as QCA simulators and offer a template for how to experimentally investigate QCA generally. We note that complex network analysis has already made a largely, though not wholly, theoretical impact on quantum information. One example arises in one-way quantum computing26 in which complex, network-structured graph states27 are irreversibly transformed using projective measurements. Another occurs in the context of the quantum internet where communication channels between geographically-distant quantum devices are either imposed28 by fiber optic networks or implied29 by satellite downlink capabilities. Notably, satellite-based quantum communication channels have recently been shown to support small-world connectivity30. Our work differs significantly from these examples. Where one-way quantum computing has experimentally realized complex graph states by design via projective measurement, our work shows that QCA dynamically generate them in an emergent fashion. Where the quantum internet considers geographic networks, our networks of correlations emerge from unitary dynamics without any notion of physical distance except the locality of interactions. Finally, the complex networks occurring herein emerge on a generally-programmable, gate-model quantum processor, that is, an experimental platform with real-world constraints, such as processor noise, not necessarily present in prior theoretical work and different in nature than in prior experimental work.

## Results

### Quantum cellular automata

A one-dimensional (1D) quantum elementary cellular automaton may be defined as a chain of L quantum bits (qubits) whose states are updated according to repeated blocks of neighborhood-local unitary operations along a discrete time axis. When every qubit’s state has been updated, a QCA cycle, t, is completed. After selecting 1D chains of high-quality qubits from the available hardware graph (Fig. 1a; see also Supplementary Note 2), the structure (Fig. 1b) of a 1D QCA experiment is comprised of an initialization step, followed by the application of some number of QCA cycle unitaries out to cycle $$t\in \{0,1,\ldots,\,{t}_{\max }\}$$, and the measurement of appropriate observables.

The particular QCA that we simulate is the totalistic, three-site Goldilocks rule T6 with a uniform Hadamard activation unitary applied to each qubit and boundary conditions equivalent to fixed $$\left|0\right\rangle$$s (see Supplementary Note 1)12. We note that the QCA notation T6 should not be confused with decoherence times, which we will denote $${\tilde{T}}_{1}$$ and $${\tilde{T}}_{2}$$ where applicable. Figure 1c shows how rule T6 is compiled down to quantum gates. A single, central bit flip initialization is followed by one QCA cycle, and finally, measurement in the computational z-basis. The local update, represented by two non-Clifford CH gates (green box), does nothing if there are zero or two adjacent $$\left|1\right\rangle$$s and applies the Hadamard activator to the central qubit if there is exactly one adjacent $$\left|1\right\rangle$$: this is the trade-off rule that gives rise to the Goldilocks nomenclature.

### Population dynamics and error mitigation

The quantum processor on which we run our QCA simulations is a 53-qubit superconducting processor, Weber, which follows the design of the Sycamore architecture outlined in ref. 20 (see also Supplementary Note 2). Typical performance characteristics for Weber are: single-qubit gate error e1 ≈ 0.1%, two-qubit gate error e2 ≈ 1.4%, $$\left|0\right\rangle$$-state readout error er0 ≈ 2%, $$\left|1\right\rangle$$-state readout error er1 ≈ 7%, and population relaxation time $${\tilde{T}}_{1}\, \approx \, 15\, \mu s$$31. Fig. 1d shows the decomposition of a single QCA cycle (red box) to the native $${\sqrt{{{\mbox{iSWAP}}}}}^{{{{\dagger}}} }$$ two-qubit and PhXZ(a, x, z) ≡ ZzZaXxZa family of single-qubit gates. Strictly speaking, the native two-qubit gate is better modeled by $${\sqrt{{{\mbox{iSWAP}}}}}^{{{{\dagger}}} }\times {{\mbox{CPHASE}}} \, (\varphi )$$, where the parasitic cphase is φ ≈ π/2321. We apply a suite of low-overhead circuit optimization, calibration, and error mitigation techniques to optimize circuit performance including moment alignment, spin-echo insertion, Floquet calibration21,24, parasitic cphase compensation, and most importantly, post-selection (see Supplementary Note 4).

At each QCA cycle depth we measure the output of the circuit in the z-basis Nc = 100, 000 times, resulting in a set of L-bit strings $$\{\left|z\right\rangle \}$$ and associated probabilities {Pz ≈ Nz/Nc}, where Nz is the number of times bit string $$\left|z\right\rangle$$ is observed. The local population on each site is calculated via $$\langle {n}_{i}\rangle=(1-{\sum }_{z}{P}_{z}{(-1)}^{{z}_{i}})/2$$ and averaged over four 1D qubit chains. Figure 2a shows the numerical emulation of such a procedure initialized with a single, central bit flip on 21 qubits out to 30 QCA cycles. The two large-scale blue diamonds indicate coherent dynamics. When repeated on the Weber processor, a combination of photon loss, gate error, and state preparation and measurement (SPAM) error leads to nearly total population decoherence by t ≈ 10 (Fig. 2b). We therefore post-select experimental data and discard any measurements whose eigenvalues of the Ising-like operator $${{{{{{{\mathcal{O}}}}}}}}=\mathop{\sum }\nolimits_{i=0}^{L}{Z}_{i}{Z}_{i+1}$$ differ from the corresponding eigenvalue of the initial state. That is, $${{{{{{{\mathcal{O}}}}}}}}$$ is a dynamical invariant of the T6 rule that keeps track of the number of domain walls in the system. Figure 2c shows that post-selection results in coherent population dynamics that persist beyond t ≈ 10, although different observables can degrade with noise on slightly different timescales (see Fig. 3). The cycle-stamped population vignettes shown in Fig. 2d support these observations more quantitatively, with error bars representing one standard deviation on the four different chains. After t ≈ 15, error bars on the post-selected data become more significant and while some qualitative features of the emulated population dynamics appear to persist, such as a larger population towards the center of the chain, it is unclear from Fig. 2 alone as to what the underlying nature of these qualitative similarities is. Moreover, our complex network analysis of the behavior of rule T6 relies on the calculation of two-body observables beyond the one-body observables depicted in Fig. 2. As such, we turn to a calculation of Shannon mutual information both to more deeply understand the long-time population dynamics of our QCA and to establish their complex network behavior.

### Mutual information network analysis

Following the complex-network approach in neuroscience wherein functional connectivity of the brain is characterized between spatially non-adjacent regions32, we calculate the classical, Shannon mutual information between all pairs of qubits in each 1D Goldilocks QCA chain

$${I}_{ij}\equiv \mathop{\sum }\limits_{{z}_{i}=0}^{1}\mathop{\sum }\limits_{{z}_{j}=0}^{1}p({z}_{i},{z}_{j}){\log }_{2}\frac{p({z}_{i},{z}_{j})}{p({z}_{i})p({z}_{j})}$$
(1)

and regard it as an adjacency matrix of correlations that defines the QCA network at each cycle. We choose to use classical, rather than a measure of quantum, mutual information because its calculation only requires measurements in the computational z-basis, which we have shown are amenable to post-selection. Moreover, we show in Supplementary Note 5 that the Shannon (classical) mutual information acts as a reliable proxy for von Neumann (quantum) mutual information for the T6 QCA.

Complex networks are ones that are neither purely regular, such as a lattice or complete graph, nor entirely random33. The classic demonstration that a network has complex, small-world character involves showing persistently large clustering and simultaneously short path length34, with a power-law node strength distribution resulting in highly-connected nodes. By analogy with transportation networks, these features describe networks that are easily traversed both locally (clustering) and globally (path length), and exhibit hubs (broad node strength distribution). While transportation networks provide an intuitive interpretation of these complex network measures, we emphasize that clustering, path length, and node strength distribution have seen widespread use in analyzing the structure of mutual information networks and in drawing conclusions regarding the physical complexity of the underlying system in both classical and quantum systems. For example, applying these measures to mutual information networks derived from electroencephalographic or fMRI data has been used to elucidate structure-function correlations in the brain32,35. In addition, the measures have been used along with earthquake time series-derived mutual information to model seismicity36 and with mutual information in wireless networks to explain routing efficiency37. Finally, complex network measures calculated on quantum mutual information networks in quantum Ising and Bose–Hubbard models are able to detect quantum phase transition critical points38. Hence, the use of clustering, path length, and node strength distribution in conjunction with mutual information is a well-established, quantitative procedure with predictive power. We employ this procedure in order to understand the structure of correlations in our QCA circuits and to observe the emergence of physical complexity in the presence of quantum processor noise.

Through decoherence, the state of a quantum processor approaches an incoherent uniformly random state with all amplitudes equal to 2L/2 and for which Iij = 0 for all L ≥ 2. Hence, the incoherent uniformly random state is neither locally nor globally traversable and is thus not a typical random network. It is also useful to consider the effect of subjecting the incoherent uniformly random state to the same post-selection procedure as our experimental data. This can be done in three steps: (i) form an incoherent uniformly random state with all 2L amplitudes equal to 2L/2, (ii) set any basis state amplitudes to zero if the basis state has an eigenvalue under $${{{{{{{\mathcal{O}}}}}}}}$$ that differs from that of the QCA’s initial condition, and (iii) renormalize the remaining amplitudes so that the state vector has unit norm. Upon post-selection, this decohered state is no longer uniform and the corresponding mutual information network is both non-zero and non-random, although to a much lesser extent than the states generated by Goldilocks QCA. The complexity of networks generated by Goldilocks QCA is established by computing network measures for emulated, raw processor, and post-selected processor states and then comparing each of these to post-selected incoherent uniform random states.

Clustering measures local network transitivity and is defined as the ratio of the weighted number of closed triangles in the network to the weighted total number of length-2 paths in the network (i.e., the number of potentially closed triangles—see Supplementary Note 6). The first relevant signature of network complexity is intermediate to large clustering values that do not decay with system size, in contrast to random networks. The emulated clustering (blue curves) of the QCA exhibits this signature and actually increases slightly as a function of system size, L (see Fig. 3). While we plot three of the larger system sizes we simulated here, L = 15, 17, and 19, this proves true for all other system sizes simulated as well. Next, we note that without post-selection, the clustering $${{{{{{{\mathcal{C}}}}}}}}$$ calculated from raw data from Weber (red points) rises briefly but then quickly decays toward zero, the incoherent uniformly random limit (black dotted curve), at t ≈ 12 for all three system sizes. In contrast, the green curves in Fig. 3 show that with post-selection the experimental clustering tracks the emulated clustering closely until t ≈ 6 and remains larger than post-selected uniform randomness (black dashed curve) until t ≈ 12, independent of system size. There is therefore a window between t ≈ 4 and 12 over which we can analyze the formation of a non-random complex network in the QCA for all system sizes simulated. We provide a more detailed description of our cycle windowing process in Supplementary Note 7.

Figure 4a shows the coherence window, cycle-and-chain-averaged emulated (blue), raw (red), and post-selected (green) clustering coefficient for L = 5–23 qubits. After the finite-size effects encountered for L ≤ 11, it is clear that while the raw clustering trends towards zero—that of an incoherent uniformly random state network—both the emulated and post-selected clustering stabilize towards $${{{{{{{\mathcal{C}}}}}}}}\,\approx\,0.3$$ and trend towards larger values as a function of system size, indicating substantial network transitivity beyond post-selected randomness (black dashed curve). Figure 4b shows the coherence window, cycle-and-chain-averaged weighted shortest path length, , as a function of system size, which gauges global network traversability (see Supplementary Notes 6 and 7). The raw data path length (red) in Fig. 4b is large and increases as a function of system size. The post-selected (green) path length tracks the emulated (blue) path length closely, trends downward, and is always one to two orders of magnitude smaller than the raw path length. Interestingly, post-selected path length tracks the path length of post-selected randomness (black dashed line) nearly as well as it does emulated path length. Taken together, however, Fig. 4a, b signal the existence of small-world mutual information networks generated in the coherence window of the Goldilocks QCA beyond what can be obtained by post-selecting incoherent uniform randomness. The end of the coherence window (t = 12) for the largest system size simulated (L = 23) corresponds to $$1056\,{\sqrt{{{\mbox{iSWAP}}}}}^{{{{\dagger}}} }$$ gates.

Figure 4c further indicates the formation of small-world mutual information networks, showing that the size-normalized emulated (blue) and post-selected (green) node strength distributions (see Supplementary Note 6), P[gi/(L − 1)], are relatively flat between 1 × 10−2 and 2 × 10−1 compared with those of the post-selected random node strengths (black dashes), which peak between ~2.5 × 10−2 and 1.5 × 10−1, and raw (red) node strengths, which are heavily biased towards much smaller values, indicating a deficit in network connectivity. Finally, Fig. 4d–g visually depicts how the mutual information for L = 23 at t = 9 differs for the raw QCA data, which approaches the network structure of incoherent uniform randomness, and the emulated and post-selected QCA networks, which both display lattice girder-like small-world structure that resemble one another more closely than they do post-selected randomness.

### Towards beyond-classical QCA

In addition to their intrinsic scientific value as quantum models for emergent complexity, QCA also present intriguing prospects for establishing new inroads into the beyond-classical era. In the instance of Goldilocks rule T6, identification of a dynamical invariant makes simulation less fragile to noise than a fully chaotic random quantum circuit (RQC). However, it also implies efficient—that is, polynomial—classical emulation via direct Schrödinger evolution since it reduces the bounded size of the protected T6 Hilbert space to scale as ~0.63 L1.91 for a fixed, single initial bit flip at the center of the chain (see Supplementary Note 4). Generally, fixing the number of initial $$\left|1\right\rangle$$s while increasing system size leads to polynomial space and time complexity. However, one can recover exponential scaling in the protected Hilbert space by simulating increasingly large chains while fixing the density (rather than number) of initial $$\left|1\right\rangle$$s well separated by $$\left|0\right\rangle$$s. For a fixed density, $${\rho }_{\left|1\right\rangle {{{{{{{\rm{s}}}}}}}}}$$, the protected Hilbert space bound scales as the binomial coefficient, $$\sim {\!}\big({{L}\atop{{\rho }_{\left|1\right\rangle {{{{{{{\rm{s}}}}}}}}}L}}\big)$$, which asymptotically scales exponentially in L. For context, the long-time, cycle-averaged bond entropy of rule T6 was shown to scale between a 1D area and volume law12. Although simulation of Goldilocks rules has shown the failure of direct matrix-product-state approaches39, given this intermediate scaling it is an open problem as to whether efficient classical simulatability may be achieved using a modified tensor network approach in the absence of a dynamical invariant12,40. Moreover, efficient simulatability of area law-scaling states in two-dimensions (2D) or higher using tensor network approaches, while promising, is even less assured than in 1D. Hence, 2D QCA that exhibit area-law scaling or worse may be good candidates for beyond-classical demonstrations.

Here we have demonstrated that existing quantum processors can simulate 1D QCA with high fidelity at large gate volume. While reliant on the availability of high-fidelity hardware, the main circuit design principles that enable this goal consist of the identification of particular QCA rules that: (i) generate significant complexity signatures, (ii) efficiently compile to low-depth sequences of hardware-native gates, and (iii) are amenable to post-selection through identification of one or more dynamical invariants (For a more complete discussion of the quantum-simulational complexity of the T6 rule including post-selection overhead, please see Supplementary Note 4). We therefore expect these design principles to aid in discovering QCA that support beyond-classical demonstrations or are otherwise useful in quantum computational domain applications. In particular, employing such design principles for QCAs that model correlated quantum matter could be a promising route toward beyond-classical simulation of novel physical systems in the near term.