Naoki Yamamoto

Topological data analysis (TDA) extracts global features of data via Betti numbers, but classical computation becomes costly for large simplicial complexes. We propose a scalable quantum TDA algorithm for Betti number estimation based on a block-encoded Dirac operator and logarithmic-depth circuit construction using reconfigurable beam splitter architectures. The method combines shallow Gaussian-Chebyshev spectral filtering with resource-focused complexity analysis. We derive error bounds, estimate gate and qubit requirements, and numerically validate correct Betti number estimation on small instances. Our results indicate that larger instances remain challenging at present but continued progress in quantum hardware may enable increasingly larger problem sizes to become realistic targets in the future.

Time-dependent Hamiltonian simulation is one of the most important applications of quantum computing, but existing approaches based on Dyson-series or Magnus-expansion constructions can require complicated circuits and large hardware overhead. In this work, we propose a cost-aware variant of continuous-qDrift that incorporates implementation cost into the sampling probabilities through importance sampling. We prove that, under the same diamond-norm error requirement as standard continuous-qDrift, the expected total cost of the cost-weighted algorithm is no larger than that of the original method. To assess practical performance, we numerically simulate a Floquet-driven Rydberg system with multi-body interaction terms and use CNOT counts as the cost metric. The results show that the proposed method achieves a similar error distribution to standard continuous-qDrift while significantly reducing total cost.

We examine photonic how to implement quantum reservoir computing with stochastic internal states on existing hardware, using a classical simulator imitating Xanadu’s photonic quantum computer, Aurora. The proposed model extracts reservoir features from measurement probabilities and modifies Aurora’s native circuit by introducing probabilistic routing between squeezed coherent and squeezed vacuum states. The framework is supported by the universality of the stochastic reservoir map, which can approximate polynomial and more general continuous functions on compact domains. In benchmark simulations, for MNIST classification with reservoir size 1200, the quantum reservoir achieved 91.4% accuracy, slightly outperforming a same-size classical echo state network. The results also suggest that readout stochasticity can improve generalization by acting as an effective regularizer.

Quantum topological data analysis is one of the most promising areas for quantum advantage. In particular, the LGZ algorithm is the most basic algorithm for estimating Betti numbers, which are considered to be the essential shape of point clouds. Since LGZ, various quantum algorithms have been proposed, but many of them are based on FTQC and require a large number of quantum resources. In this paper, we propose a QTDA algorithm that does not require ancilla bits using a Hamiltonian based on supersymmetry, which has not been widely used in existing research. In particular, by imposing constraints on the graph, we show that there is always a known 0 eigenvalue and eigenstate in the Hamiltonian, this allows us to use ancilla-free measurement algorithms. Moreover, by using randomization, we reduce the required number of samples.

Density matrix exponentiation (DME) is a general procedure that converts an unknown quantum state into the Hamiltonian evolution. This enables state-dependent operations and can reveal nontrivial properties of the state, among other applications, without full tomography. However, it has been proven that for any physical process, the DME requires $\Theta(1/\varepsilon)$ state copies in error $\varepsilon$. In this work, we go beyond the lower bound and propose a procedure called the \textit{virtual} DME that achieves $\mathcal{O}(\log(1/\varepsilon))$ or $\mathcal{O}(1)$ state copies, by using non-physical processes. Using the virtual DME in place of its conventional counterpart realizes a general-purpose quantum algorithm for property estimation, that achieves \textit{exponential} circuit-depth reductions over existing protocols across tasks including quantum principal component analysis, quantum emulator, calculation of nonlinear functions such as entropy, and linear system solver with quantum precomputation. In such quantum algorithms, the non-physical process for virtual DME can be effectively simulated via simple classical post-processing while retaining a near-unity measurement overhead. We numerically verify this small constant overhead together with the exponential reduction of copy count in the quantum principal component analysis task. The number of state copies used in our algorithm essentially saturates the theoretical lower bound we proved.

Circuit knitting/cutting is a family of techniques that enables large quantum computations on limited-size quantum devices by decomposing a target circuit into smaller subcircuits. However, it typically incurs a measurement overhead exponential in the number of cut locations, and this scaling has long been believed to be fundamentally unavoidable. In this work, we show that such an exponential scaling is not universal: it can be circumvented for tree-structured quantum circuits via concatenated quantum tomography protocols. We first consider the task of estimating the expectation value of an observable within additive error $\epsilon$ for a tree-structured circuit with tree depth 1, maximum branching factor $R$, and bond dimension at most $d$ on each edge. Our approach uses quantum tomography to construct, for each cut edge, a local decomposition that eliminates the rescaling factors in conventional QPD, instead introducing a controllable bias set by the tomography sample size. As a result, we show that $\mathcal{O}(d^3R^3\ln(dR)/\epsilon^2)$ total measurements suffice, including tomography measurements. Next, we extend the tree-depth-1 case to general trees of depth $L\geq2$, and give an algorithm whose total measurement cost $\mathrm{poly}(d,K,1/\epsilon)$ scales polynomially with the number of cuts $K$ for complete multi-ary trees. Finally, we perform an information-theoretic analysis to show that, in a comparable tree-depth-1 setting, conventional circuit-cutting methods require at least $\Omega((d+1)^R/\epsilon^2)$ measurements. This exponential separation in the number of cuts suggests that the improvement is not solely due to the tree restriction, highlighting the essential role of tomography-based construction in reducing measurement overhead in hybrid quantum–classical computations.

Preparing low-energy eigenstates, such as excited states, in quantum systems is a central challenge. Previous method reduces this to a ground-state preparation problem by modifying the system Hamiltonian using an approximate energy of the target state. However, this approach results in a large Hamiltonian simulation cost. Here, by adding a penalty term based on a system-size-independent conserved quantity, we successfully reduce this cost by a square-root factor. We confirm this through numerical simulations on the transverse-field Ising model and the Hubbard model.