Quantum excited states are the hidden chapters of nature’s story, the spectral fingerprints that light up when molecules vibrate, electrons hop, or spins flip. They’re essential to understanding chemistry, materials, and even how we design quantum devices. Yet for all the fuss around quantum computing and advanced simulations, predicting those excited states remains stubbornly hard. Ground-state methods are well-tuned, but once you venture into the realm of the first few excited levels, the math and the bookkeeping get tangled in orthogonality constraints, penalties, and a cycle of state-specific optimizations. A new study from the Institute of Physics, Chinese Academy of Sciences in Beijing, led by Shi-Xin Zhang and Lei Wang, proposes a bold twist: instead of wrestling with orthogonality one state at a time, build a small, non-orthogonal family of trial states and optimize the subspace they span. The payoff is a unified variational principle that can reveal multiple low-energy excited states simultaneously.
Think of it as conducting a chorus rather than training soloists. You keep a handful of candidate wavefunctions, let them mingle and mix, and then you pull out the best harmonies—the lowest energies—without forcing any one voice to stay perfectly apart from the rest during the rehearsal. The trick is mathematical elegance: the method minimizes the trace of the inverse overlap matrix times the Hamiltonian, L = Tr(S^{-1} H), where S is the overlap matrix of non-orthogonal trial states and H is the system’s energy operator. The trick also anchors itself in a time-honored principle from linear algebra, the Ritz or Rayleigh-Ritz approach, but it brings it into a form that’s friendly to modern optimization tools and a variety of wavefunction representations. The result is a flexible, end-to-end framework that can sit atop classical simulators like matrix product states and tensor trains or run inside variational quantum circuits on quantum hardware.
A Unified Variational Frame for Excited States
At the heart of the method are Ns trial states |ψi⟩, not required to be normalized or orthogonal. From them one builds the Ns×Ns overlap matrix Sij = ⟨ψi|ψj⟩ and the Hamiltonian matrix Hij = ⟨ψi|H|ψj⟩. The leap is to minimize L(θ) = Tr[S(θ)^{-1} H(θ)], where θ stands for all variational parameters indexing the Ns states. Because S is Hermitian and positive definite (as long as the |ψi⟩ are linearly independent), its inverse exists and the trace captures the sum of the approximate Ritz energies in the current subspace. In other words, you’re deliberately shaping the subspace to pull in the lowest possible sum of energy levels, not chasing a single ground state with a single-trajectory optimization.
The theoretical backbone rests on the generalized eigenvalue problem Hc = ESc. Once the variational parameters have steered the subspace into the right neighborhood, solving this small, Ns×Ns problem recovers the approximate eigenstates |Ψα⟩ = ∑i ciα|ψi⟩ and their energies Eα. A key advantage is that you don’t need explicit orthogonality constraints or penalty terms during the optimization loop. You optimize a single scalar loss with standard gradient-based methods, and the orthogonal structure appears naturally in the post-processing step when you diagonalize the Hamiltonian within the spanned subspace. This separation—optimize the subspace, then diagonalize—keeps the training landscape smoother and leverages the strength of automatic differentiation.
To the authors, the move is practical as well as principled. The gradient of L with respect to the variational parameters can be computed automatically, unlocking the same kinds of hardware-accelerated workflows that power training large neural networks. And because the inverse overlap S^{-1} only acts on an Ns×Ns matrix, the computational footprint for the core optimization scales with the number of excited states you want, not with the size of the full quantum system. The authors emphasize that their approach is agnostic to the choice of ansatz. You can use matrix product states, tensor trains, or variational quantum circuits—whatever fits the problem and the hardware best. The method acts as a unifying scaffold for excited-state calculations across classical and quantum platforms.
As a note of context, the work sits at the intersection of several long-running threads in quantum many-body physics: how to represent complex wavefunctions efficiently, how to extract spectral information without grueling iterative diagonalizations, and how to bridge classical and quantum computational paradigms. The paper’s authors explicitly connect their loss to the Ritz values through the minimax principle, showing that the approach yields Ritz upper bounds to the exact excited energies. In practice, that translates to reliable, convergent estimates for the first several excited levels, provided the chosen variational space is expressive enough. The collaboration explicitly demonstrates this with three distinct testbeds, each showcasing a different flavor of variational representation.
Three Experiments Across Quantum Realms
First, the method tackles a classic quantum many-body problem: a one-dimensional spin-1/2 chain with periodic boundary conditions. Using a set of Ns = 32 independent matrix product states (MPS) with a bond dimension χ = 16, the researchers target the lowest 32 energy levels. The state representations live on a ring of N = 16 sites, a nontrivial setting that tests how the approach handles finite-size effects and nonlocal correlations. As the optimization progresses, the loss L = Tr[S^{-1} H] decreases steadily, signaling that the variational states are filling out the subspace that captures the low-energy spectrum. When they diagonalize H within this subspace, the resulting energies match the exact spectrum with extraordinary fidelity, and the reported relative errors hover around 10^{-7} across the hierarchy of levels. It’s a clean demonstration that a non-orthogonal ensemble of trial states can unlock a faithful multi-state picture without resorting to penalty terms or sequential hunts for each excited state.
Second, they push into a continuous-variable setting: the Morse potential, a textbook model for diatomic vibrations, discretized on a fine real-space grid. Here the wavefunction lives on a grid space and the excited vibrational levels are sought simultaneously using a quantics tensor train (TT) representation—an efficient matrix-product-like format for continuous coordinates. The TT wavefunctions are parameterized by a network of cores with a sizable TT rank χ = 128, enabling a faithful capture of the intricate wavefunctions of several vibrational states. Targeting the lowest Ns = 16 vibrational levels, the method reproduces the known spectrum with relative accuracy on the order of 10^{-5}. This is significant because it shows the framework’s versatility beyond discrete lattice models: continuous spectra, often relevant for spectroscopy, also fall under the same umbrella when you encode the wavefunction in a TT/Matrix Product form. The convergence of the loss curve in this setting mirrors the spin-chain story: the subspace is being tuned to cradle the right set of low-energy excitations.
Third, the authors move to a more strongly correlated electronic system: a two-dimensional Hubbard model on a 2 × 3 lattice, mapped to 12 qubits via a Jordan-Wigner transformation. They deploy Ns = 16 independent variational quantum circuits, each a hardware-efficient ansatz built from five repeating blocks. The objective is again the sum of low-energy Ritz values, but now the states are prepared by quantum hardware-inspired circuits, with the overlaps and Hamiltonian matrix elements estimated on a quantum device or simulator. They benchmark against a subspace variational quantum eigensolver (VQE) approach that constructs a static subspace and then diagonalizes within it. Across ten independent optimization trials, their method consistently achieves a smaller loss and a tighter spread, with the extracted excited energies aligning closely with the expected values. The comparison underscores a practical advantage: by optimizing the subspace itself rather than chasing a single ground-state energy, the method yields a more robust and potentially more scalable path to multiple low-energy excitations on quantum hardware.
Across these three stories, a throughline emerges: you can rack up a handful of non-orthogonal, parameterized states, measure the right cross-terms between them, and let a single loss function shepherd the ensemble toward the physics you care about. The final extraction of the energies and states uses a standard generalized eigenproblem, but the heavy lifting—finding the right subspace—happens in the differentiable optimization loop. In every setting, even with different representations (MPS, TT, PQCs), the framework lands on the low-energy spectrum with high fidelity. The training dynamics may be nontrivial and the optimization landscape non-convex, but the empirical results suggest a robust path to simultaneously capture multiple excited states without the usual penalties or cumbersome, state-by-state tuning.
Why It Matters for Science and Technology
So what does this matter beyond a clever theoretical trick? Quite a lot, actually. Excited states govern spectroscopy—the fingerprints scientists use to infer bond strengths, reaction pathways, and material properties. Having a unified, flexible method to compute several of these states in one go could dramatically accelerate how chemists and physicists screen molecules, design catalysts, or predict spectral lines. The approach is deliberately agnostic about the choice of wavefunction representation. That means it can ride on powerful classical simulators—MPS for quasi-one-dimensional systems, TT for higher-dimensional or continuous problems—or be embedded in the growing ecosystem of variational quantum algorithms. In other words, the same mathematical idea scales across platforms, letting researchers pick the most practical tool for the problem at hand.
The method also plays nicely with the broader wave of differentiable programming sweeping physics, chemistry, and machine learning. Because the loss is differentiable with respect to the variational parameters, you can plug in modern automatic-differentiation pipelines, backpropagation-friendly optimizers, and even neural quantum states if you want to push expressive power further. When you couple that with the insight that you’re optimizing a subspace rather than fighting orthogonality term-by-term, you gain a more forgiving training landscape. This could be pivotal for tackling larger, more complex systems where penalties and penalties-only formulations become brittle or unstable. The paper’s authors even note the potential to blend this framework with neural-network quantum states, giving a path toward hybrid quantum-classical workflows that can roam across regimes—from strongly correlated electrons to molecular vibrations.
There’s also a pragmatic elegance to the scaling argument. S, the Ns×Ns overlap matrix, stays small by design—the number of states you want to resolve, not the size of the full Hilbert space. That means the expensive bit, in many cases, is still building and evaluating the matrix elements Hij and Sij for the chosen ansatz, not solving an enormous eigenproblem inside a gigantically dimensional space. In the world of quantum simulation, where every extra qubit or every extra tensor dimension carries a cost, this separation of concerns—optimize a compact subspace, then diagonalize—could be a practical recipe for near-term quantum devices.
In short, Zhang and Wang offer a unifying lens: excited states aren’t a scattered set of one-off problems to be groped for with ad-hoc tricks; they’re a spectrum tied together by the subspace spanned by a carefully chosen family of trial states. If you want to map that spectrum quickly and reliably, you should design the subspace well, and then let the standard machinery of linear algebra pull out the energies. The architecture is deliberately flexible, and that may be exactly what the field needs as it experiments with new hardware, new wavefunction forms, and new applications—from quantum chemistry to condensed matter to quantum information science.
The study is a reminder that sometimes the most powerful advances come not from building a bigger hammer, but from framing the problem differently. By reframing excited-state calculation as a trace-minimization over a subspace of non-orthogonal trial states, the authors craft a tool that is at once conceptually clean and practically adaptable. If it proves robust as researchers push toward larger systems and more expressive ansatzes, we might see a wave of new multi-state spectroscopies, faster materials discovery, and smoother pathways to leveraging quantum computers for the messy, beautiful physics of excited states. The journey from theory to bench to bit may still have choppy patches, but this framework provides a steadier compass for navigating the spectrum.
Lead institutions and researchers: The work is published from the Institute of Physics, Chinese Academy of Sciences in Beijing, with Shi-Xin Zhang and Lei Wang at the helm, illustrating how ideas born in quantum many-body theory can cross-pollinate with modern machine learning and quantum hardware to illuminate the excited-manifold of quantum systems.
