Infinite temperature at zero energy

This paper presents a construction of static, geometrically local Hamiltonians with provably volume-law-entangled ground states and infinite-temperature characteristics across their entire spectrum, achieved by adapting the Feynman-Kitaev clock with periodic boundary conditions and leveraging exactly solvable Floquet circuits that satisfy the eigenstate thermalization hypothesis.

The Gist

The Big Idea: A "Time Machine" for Quantum States

Imagine you have a quantum computer running a complex program. Usually, when a quantum system gets very hot (infinite temperature), its particles become a chaotic, random mess. This is called "thermalization," and it's a state where everything is jumbled up and highly connected (entangled).

Normally, the ground state (the lowest energy, coldest state) of a physical system is very orderly. It's like a calm, frozen lake. In most physics, we expect the ground state to be simple and have very little "entanglement" (connection) between different parts of the system.

This paper does something surprising: It builds a special kind of "time machine" (a mathematical construction) that takes a chaotic, hot, random state from a quantum circuit and traps it inside the ground state of a new, static machine. The result? They created a system where the coldest, most stable state is actually a chaotic, highly connected mess.

The Two Main Ingredients

To build this, the authors combined two different tools:

1. The "Feynman-Kitaev Clock" (The Time Machine)

Think of a quantum circuit as a movie. It starts with a scene, plays through a series of actions (gates), and ends with a final scene.

• The Old Way: Usually, physicists use a "clock" to record the movie frame-by-frame. The clock starts at 0, goes to 1, 2, and stops at the end. This is an "open" clock.
• The New Way (This Paper): The authors changed the clock to be a loop. Imagine a clock hand that runs around a circular track. When it reaches the end, it instantly jumps back to the start.
• The Magic: Because the clock is a loop, the "movie" it records must also be a loop. The end of the movie must match the beginning. In physics terms, this forces the system to settle into a Floquet state (a state that repeats itself over time).
• The Result: By making the clock a loop, they forced the ground state of their new machine to contain the "memory" of a repeating quantum movie.

2. The "LFSR" (The Chaotic Generator)

To make sure the "movie" inside the loop is actually chaotic and random (and not just a boring, simple pattern), they needed a specific type of quantum circuit.

• They used a circuit based on Linear Feedback Shift Registers (LFSRs).
• The Analogy: Think of an LFSR like a very clever, deterministic slot machine. It takes a row of bits (0s and 1s), shifts them over, and uses a specific rule to generate a new bit to put at the front.
• The Trick: While this machine follows strict, simple rules (it's "solvable" and not random by accident), the patterns it generates look incredibly random to anyone watching. It's like a perfectly predictable recipe that somehow produces a dish that tastes completely unpredictable.
• The Proof: The authors mathematically proved that the states produced by this LFSR machine are "thermal." They are so scrambled that any small piece of the system looks like it's at infinite temperature.

Putting It Together: The "Hot" Ground State

When the authors combined the Loop Clock with the Chaotic LFSR, they created a new physical system (a Hamiltonian).

• The Ground State: Usually, the ground state is the "quietest" state. But in their system, the ground state is built from the chaotic LFSR states.
• The Entanglement: In physics, "entanglement" is a measure of how connected different parts of a system are.
  • Normal Ground State: Low entanglement. If you cut the system in half, the two halves barely know about each other. (This is called an "Area Law").
  • Their Ground State: High entanglement. If you cut the system in half, the two halves are deeply connected, just like they would be in a super-hot, chaotic gas. (This is called a "Volume Law").

The Key Achievement: They proved that every single piece of their system, no matter how small or where it is, is highly entangled. This is different from previous examples of "hot ground states," which were only entangled in a specific, structured way (like a rainbow pattern). Their system is entangled everywhere, looking truly random.

Why Is This Important?

1. Breaking the Rules: It challenges the old belief that cold, stable states must be simple and orderly. It shows that you can have a "frozen" state that is actually a "scrambled" mess.
2. A New Tool for Study: They created a specific, solvable model (the LFSR circuit) that behaves like a chaotic, thermal system. This is rare. Usually, proving a system is chaotic is very hard. This gives scientists a clean, mathematical "test bed" to study how quantum systems heat up and become random.
3. The "Edge Case": The system behaves strangely. It is chaotic enough to look thermal, but because it's built on simple rules, it doesn't behave exactly like a truly random system. It's a unique "edge case" that helps physicists understand the difference between "looking random" and "being random."

Summary Analogy

Imagine a library (the quantum system).

• Normal Physics: The "Ground State" is the quietest, most organized section of the library. Books are perfectly sorted, and no one is talking.
• This Paper: The authors built a special library where the "Ground State" is a section where a chaotic, loud party is happening 24/7. Everyone is shouting, books are flying everywhere, and the noise level is at maximum.
• The Twist: Even though it looks like a chaotic party, the rules of the library are so strict and simple that a mathematician could predict exactly where every book will be next second. It's a predictable chaos that lives in the coldest, most stable part of the building.

This paper proves such a library can exist and describes exactly how to build it.

Technical Summary

Technical Summary: Infinite Temperature at Zero Energy

Problem Statement
In many-body physics, the spatial scaling of entanglement entropy serves as a primary diagnostic for distinguishing quantum phases. High-energy eigenstates of generic interacting systems typically obey the Eigenstate Thermalization Hypothesis (ETH), exhibiting "volume-law" entanglement where entropy scales with the subsystem volume. Conversely, ground states of local Hamiltonians generally obey an "area law" (scaling with boundary size) or, in gapless critical systems, a logarithmic scaling. While previous constructions (e.g., Motzkin and Fredkin chains) demonstrated volume-law entangled ground states in one dimension, these states possess a distinct "rainbow" structure: entanglement is dominated by long-range correlations between symmetric pairs of sites, failing to exhibit volume-law scaling for arbitrary contiguous local subsystems. The central open question addressed by this work is whether a local, static Hamiltonian can possess a ground state that is truly thermal-like—obeying volume-law entanglement for all contiguous subsystems, mimicking the random, featureless character of infinite-temperature states.

Methodology
The authors employ a two-pronged construction combining a modified Feynman-Kitaev (FK) clock model with a specific class of exactly solvable Floquet circuits.

1. Periodic FK Clock Model: The standard FK clock maps the output of a quantum circuit to the ground state of a static Hamiltonian. The authors modify this by imposing periodic boundary conditions on the clock register. Instead of an open chain where time evolves from $t=0$ to $T-1$, the clock is a ring where the final state connects back to the initial state.

   • The Hamiltonian acts on a "two-leg ladder" consisting of a spin chain (qubits) and a clock chain (fermions).
   • The clock particle (fermion) hops around the ring, and its position dictates which unitary gate from an input circuit is applied to the spins.
   • This construction embeds the Floquet eigenstates (eigenstates of the periodic time-evolution operator $U_{T:0}$) of the input circuit into the eigenstates of the static Hamiltonian.
   • Crucially, the authors prove that if the input circuit's Floquet states obey the ETH at infinite temperature, the resulting static Hamiltonian's eigenstates (including the ground state) inherit volume-law entanglement for all contiguous subsystems.

2. LFSR Floquet Circuits: To avoid relying on the unproven assumption that generic circuits obey ETH, the authors construct a specific family of Floquet circuits based on Linear Feedback Shift Registers (LFSRs).

   • These are classical reversible circuits implemented as quantum "staircase" circuits using SWAP and CNOT gates.
   • The authors rigorously prove that the non-trivial eigenstates of maximal LFSR circuits obey the ETH at infinite temperature. They demonstrate that matrix elements of local Pauli operators between these eigenstates are exponentially small and follow a distribution consistent with pseudorandomness, despite the circuits being Clifford circuits (which do not generate "magic" or non-stabilizer resources).

Key Contributions and Results

• Theoretical Construction of Volume-Law Ground States: By combining the periodic FK clock with LFSR circuits, the authors provide the first rigorous construction of a static, geometrically local 1D Hamiltonian whose ground state is provably volume-law entangled for all contiguous subsystems up to half the system size. This distinguishes the result from prior "rainbow" states, where local subsystems could have low entanglement.
• Proof of ETH for LFSR Eigenstates: The paper establishes that maximal LFSR circuits satisfy the diagonal ETH at infinite temperature. This is achieved by bounding the matrix elements of Pauli operators using tools from algebraic number theory (mixed-character sums), showing they behave like random states despite the circuits' highly structured, non-chaotic nature (in terms of magic generation).
• Spectral Properties: The resulting Hamiltonian has a bounded spectrum (energy range $[0, 2]$) and an exponentially small gap above the ground state ($O(4^{-n})$). The ground state can be tuned to be frustration-free by introducing a specific phase twist (flux) in the clock boundary conditions.
• Infinite-Temperature Observables: The ground state exhibits a unique duality: while the clock sector behaves like a zero-temperature free-fermion system, the spin sector behaves as if it is at infinite temperature. Local observables supported solely on the spins have expectation values consistent with the infinite-temperature ensemble (maximally mixed reduced density matrices), even though the global state is a zero-energy eigenstate.

Significance and Claims
The authors claim this work provides a new class of exactly solvable models that serve as "edge cases" for understanding quantum chaos, thermalization, and entanglement.

• Challenging Conventions: The construction demonstrates that ground states can possess entanglement structures indistinguishable from highly excited thermal states for all local measurements, challenging the belief that low-energy states must have limited correlations or specific structural constraints (like the rainbow state).
• Decoupling Chaos Criteria: The LFSR circuits serve as a counterexample to the conflation of different chaos criteria. They satisfy ETH (thermalization of local observables) and have maximal Krylov complexity, yet they are Clifford circuits (generating no magic) and have evenly spaced spectra (violating random matrix statistics and resonance conditions). This clarifies that ETH does not necessarily imply random-matrix-like spectral statistics or the generation of non-stabilizer resources.
• Complexity Implications: The paper notes that if Floquet states with super-polynomial circuit complexity exist, the FK clock construction implies that static local Hamiltonian ground states can also possess super-polynomial circuit complexity, potentially challenging the efficiency of Matrix Product State (MPS) representations for 1D ground states.
• Stability: The authors modestly acknowledge that the specific volume-law ground states are fragile to generic perturbations (likely opening a gap), but the class of Hamiltonians with infinite-temperature properties is extensive. They suggest the model is useful for studying the interplay between driven (Floquet) physics and static Hamiltonians, and for mapping Floquet state preparation problems to ground-state search problems.

The work does not propose experimental realizations or specific applications beyond theoretical insights into many-body dynamics and quantum information complexity.