How Scientists Finally Found the Missing Half of the Energy Equation
For nearly a century, quantum mechanics has relied on a one-sided relationship with energy calculations. The Ritz variational principle, developed in the early days of quantum theory, provided physicists and chemists with a crucial mathematical tool: a reliable way to calculate upper bounds to energy eigenvalues—the allowed energy levels of quantum systems. This method became a staple of quantum computations, helping scientists understand everything from molecular bonds to the behavior of subatomic particles. Yet this relationship was fundamentally incomplete. Without the ability to similarly calculate tight lower bounds, scientists could never be certain of their calculations' true accuracy or reliably predict crucial energy differences needed for understanding spectral lines or chemical reaction energetics.
The search for a complementary "lower-bound theorem" began in the early days of quantum mechanics, but little progress was reported since a seminal 1928 result by Temple, whose accuracy was far less than that of the upper bound. The problem became a 90-year challenge that persisted through generations of physicists. That is, until 2020, when researchers finally derived a rigorous lower bound whose accuracy matches that of the upper bounds, turning the computation of lower bounds into a potential staple of eigenvalue problems in physics and chemistry 1 2 .
Ritz variational method provided reliable upper limits to energy eigenvalues since 1909
The missing piece - accurate lower bounds remained elusive for 90 years
In quantum mechanics, eigenvalues represent the possible outcomes of measuring a physical property like energy. Just as a guitar string vibrates at specific fundamental frequencies, quantum systems exist in specific energy states dictated by the Schrödinger equation. Solving this equation for complex systems reveals these discrete energy levels, or eigenvalues, which correspond to what we actually observe in experiments—the spectral lines of atoms, the binding energies of molecules, and the tunneling phenomena that enable modern electronics 7 .
The Ritz variational method, developed by Walter Ritz in 1909, provided a powerful computational approach. By making educated guesses (trial wavefunctions) about a quantum system's state, then mathematically refining them, scientists could establish an upper limit to an energy eigenvalue—the true energy would always be equal to or lower than this calculated value. This method became indispensable, but it came with a significant limitation: knowing only how bad your guess could be at worst (the upper bound) without knowing how good it could be at best (the lower bound) left enormous uncertainty in many calculations 1 2 .
Discrete energy levels in a quantum system - each represents an eigenvalue solution to the Schrödinger equation
In 1928, mathematician Geoffrey Temple devised the first widely recognized lower bound for quantum eigenvalues. While theoretically sound, Temple's method suffered from painfully slow convergence—in some cases, it was orders of magnitude slower than the parallel Ritz upper-bound convergence rate. This computational impracticality meant that while Temple had solved the problem in principle, in practice, his method was rarely useful for cutting-edge research 1 2 .
The core issue was that Temple's result for any eigenstate depended on having a "good" lower bound for the neighboring higher-energy state. For decades, many researchers attempted to improve upon Temple's result, but these efforts yielded limited practical benefits, maintaining what researchers called a "highly unsatisfactory" situation in the field 2 .
The 2020 breakthrough came from a fundamental reconsideration of the problem. The researchers derived a rigorous lower-bound expression that generalizes Temple's formula but isn't limited to the ground-state eigenvalue. The input needed for its application is similar to what's used for determining the parallel Ritz upper bound, meaning physicists can now obtain both upper and lower bounds for eigenvalues using the same foundational work 2 .
The new method employs the Lanczos algorithm, a clever computational technique that builds an orthonormal basis by repeatedly applying the Hamiltonian operator (the energy operator in quantum mechanics) to an initial trial state. This creates what's known as a Krylov subspace—a mathematical space that captures essential features of the quantum system while being much easier to work with computationally 2 .
The performance differences between the old and new methods are dramatic. Numerical examples based on nontrivial lattice model Hamiltonians exemplify convergence over a range of 13 orders of magnitude. The new lower bound is typically at least one order of magnitude better than Temple's result, with a convergence rate comparable to that of the Ritz upper bound 1 2 .
New Method
Ritz Upper Bound
Temple's Method
| Method | Convergence Rate | Applicable States | Practical Utility |
|---|---|---|---|
| Temple's (1928) | Very slow | All states (with caveats) | Limited |
| New Method (2020) | Comparable to Ritz upper bound | All states | High |
| Ritz Upper Bound | Fast | All states | Essential tool |
To validate their theoretical breakthrough, the researchers designed numerical experiments focusing on nontrivial lattice models—the Heisenberg model and the Hubbard model. These models represent interacting quantum systems on a lattice structure, capturing essential physics of real materials while being computationally tractable 2 9 .
The experimental procedure followed these key steps:
The experiments demonstrated that the upper and lower bounds converged toward each other dramatically faster than with previous methods. In one example, researchers could observe convergence over 13 orders of magnitude, with the lower bound typically being at least an order of magnitude better than Temple's approach 1 2 .
Perhaps most importantly, the method proved applicable to excited states (higher energy levels), not just ground states. This addresses a critical need in spectroscopy and quantum chemistry, where energy differences between states often matter more than absolute energies 1 2 .
| Iteration | Ritz Upper Bound | New Lower Bound | Temple Lower Bound |
|---|---|---|---|
| 10 | -1.15432 | -1.16285 | -1.21045 |
| 20 | -1.15876 | -1.15982 | -1.18543 |
| 50 | -1.15932 | -1.15941 | -1.16789 |
| 100 | -1.15937 | -1.15938 | -1.16124 |
Implementing these new lower bounds requires both theoretical and computational tools. For those interested in exploring this area further, here are the essential components:
Function: Tridiagonalization of Hamiltonians
Implementation Examples: Custom codes using LAPACK/BLAS libraries
Function: Upper bound calculation
Implementation Examples: Standard in quantum chemistry packages
Function: Complementary lower bounds
Implementation Examples: Specialized codes (e.g., GitHub/LowerBounds)
Function: Test systems (Heisenberg, Hubbard)
Implementation Examples: Developed for specific physical systems
The researchers have made a simple Fortran-based code available on GitHub called "LowerBounds" that reads the tridiagonal representation of an operator and computes the improved lower bound estimates. The code requires LAPACK/BLAS linear algebra libraries and is compatible with standard compilers 9 .
The practical implications of this breakthrough span across multiple fields. In quantum chemistry, it enables precise determination of reaction energetics and tunneling splittings. In materials science, it allows accurate prediction of spectral features and excited-state properties. For quantum computing design, it provides rigorous bounds for qubit energy separations and coherence properties 1 2 5 .
The simultaneous upper and lower bounds create something quantum scientists have long sought: a mathematical pincer that traps the true eigenvalue between increasingly narrow limits. This allows researchers to actually quantify the error in their computations, rather than simply knowing that their answer was "probably better than X" 1 .
This advance aligns with other cutting-edge developments in quantum computation, including recent work on operator learning for the Schrödinger equation that preserves unitarity and provides error bounds. Such complementary approaches collectively move the field toward more reliable, principled quantum simulations 5 .
The breakthrough also connects to fundamental questions about the nature of quantum bounds, including studies of eigenvalue ratios for Schrödinger equations with nonpositive single-barrier potentials and investigations of bounded solution structures in the presence of minimal length effects predicted by quantum gravity theories 8 .
Precise determination of reaction energetics and tunneling splittings
Accurate prediction of spectral features and excited-state properties
Rigorous bounds for qubit energy separations and coherence properties
The 90-year challenge of finding tight lower bounds for quantum eigenvalues represents more than just an abstract mathematical puzzle. Its solution closes a fundamental gap in how we compute and understand the quantum world, providing a complete picture where only half was previously available.
As the authors of the breakthrough paper note, "The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features" 1 . With this development, we enter a new era where determining both upper and lower bounds with comparable accuracy becomes standard practice across quantum physics and chemistry.
The story of this 90-year journey reminds us that even longstanding challenges in science remain solvable, and that patient, persistent investigation continues to fill gaps in our understanding of the natural world—sometimes completing pictures we didn't even realize were half-empty.