Krylov Time Reversal (KTR)

• Krylov Time Reversal (KTR) is a quantum diagonalization approach that leverages time-reversal symmetry to extract real-valued Krylov matrix elements via shallow, control-free circuits.
• The protocol eliminates the need for controlled unitaries and ancillas by measuring Pauli-string observables, thereby reducing circuit depth and experimental overhead.
• Empirical studies on models like the transverse-field Ising and Z₂ gauge-Higgs models validate KTR’s efficient spectral estimates with high accuracy and minimized errors.

Krylov Time Reversal (KTR) is a quantum diagonalization protocol designed to enable extremal spectral estimation of Hamiltonians on near-term quantum devices without the overhead of controlled-unitary operations or ancilla qubits. KTR exploits time-reversal symmetry in Hamiltonian evolution, allowing the extraction of real-valued Krylov matrix elements via shallow circuits consisting only of Pauli-string measurements. This renders the approach especially suitable for hardware-constrained architectures and models with appropriate symmetries (Mariella et al., 30 Jul 2025).

1. Classical Krylov Subspace Diagonalization and Quantum Generalization

Quantum Krylov Diagonalization (KQD) computes extremal eigenvalues of a Hamiltonian $H$ by constructing a Krylov subspace from time-evolved versions of an initial state $\ket{v_0}$: $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$ Overlap matrices,

$$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$

are assembled and the spectral problem is reduced to solving the generalized eigenvalue equation

$A\mathbf{x} = \lambda B \mathbf{x}.$

Traditional quantum KQD requires accessing complex overlaps and transition matrix elements, typically via controlled-unitary Hadamard tests, which are challenging on NISQ devices due to circuit depth and control requirements (Mariella et al., 30 Jul 2025).

2. Time-Reversal Symmetry and Algebraic Foundations of KTR

KTR removes the need for controlled circuits by exploiting a unitary time-reversal involution $T$: $T^2=I, \qquad T^\dagger=T, \qquad \{T,H\}=0.$ For an initial state $\ket{v_0}$ satisfying $T\ket{v_0}=c\ket{v_0}$ with $c\in\{\pm1\}$, $\ket{v_0}$0 reverses the sign of $\ket{v_0}$1 under conjugation. This symmetry yields the following exact relations for overlaps between time-evolved Krylov states:

• Gram overlaps (Lemma 2.1): For any $\ket{v_0}$2, with $\ket{v_0}$3,

$\ket{v_0}$4

• Hamiltonian overlaps (Lemma 2.2):

$\ket{v_0}$5

Consequently, the entire Krylov matrix pencil is constructed from expectation values of Hermitian Pauli-strings at a single intermediate time. All elements of $\ket{v_0}$6 are real-symmetric (Toeplitz), and those of $\ket{v_0}$7 are imaginary-skew-Toeplitz (Mariella et al., 30 Jul 2025).

3. Quantum-Circuit Implementation

The KTR protocol requires only shallow, control-free quantum circuits:

• Preparation: For a chosen initial state $\ket{v_0}$8, produce $\ket{v_0}$9 for each time-shift $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$0 in the selected grid.
• Measurement of $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$1:

1. Prepare $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$2.
2. Rotate each qubit into the eigenbasis of its associated Pauli in $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$3 (e.g., $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$4 if $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$5).
3. Measure all qubits in the computational basis; the product of bits yields the eigenvalue $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$6.
4. Estimate expectation over multiple shots.

• Measurement of $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$7:

1. For $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$8, decompose $$\text{Span}\{\ket{v_0}, e^{-iHt_1}\ket{v_0}, \ldots, e^{-iHt_{m-1}}\ket{v_0}\}.$$9.
2. For each Pauli-string $$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$0, execute basis rotations and measurement as above, weighting contributions by $$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$1.

No ancilla qubits or controlled unitaries are ever required. The circuit depth is typically half that of controlled-evolution Hadamard-test methods, plus a negligible overhead for basis rotations (Mariella et al., 30 Jul 2025).

4. Krylov Subspace Construction and Generalized Eigenproblem

KTR's algorithmic pipeline mirrors standard KQD with modifications for overlap measurement:

1. Choose $$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$2 so that $$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$3 and with high overlap with low-energy eigenstates of $$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$4.
2. Select a time-grid $$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$5, frequently uniform.
3. For each $$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$6,

$$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$7

1. Assemble $$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$8 (real-symmetric) and $$B_{ij} = \braket{v(t_i)}{v(t_j)}, \quad A_{ij} = \bra{v(t_i)}H\ket{v(t_j)},$$9 (imaginary-skew-symmetric), and solve $A\mathbf{x} = \lambda B \mathbf{x}.$0.
2. The smallest eigenvalues $A\mathbf{x} = \lambda B \mathbf{x}.$1 approximate extremal eigenvalues of $A\mathbf{x} = \lambda B \mathbf{x}.$2.

Explicit orthonormalization—e.g., via a Lanczos-type recurrence—can be performed, but most implementations favor direct diagonalization of the $A\mathbf{x} = \lambda B \mathbf{x}.$3 matrix pencil for efficiency. All matrix elements required are observable expectation values amenable to batching and post-processing (Mariella et al., 30 Jul 2025).

5. Empirical Benchmarks and Spectral Estimation

Numerical validation is provided via MPS-based simulation on paradigmatic models:

• Transverse-Field Ising Model (TFIM): For $A\mathbf{x} = \lambda B \mathbf{x}.$4 qubits, with $A\mathbf{x} = \lambda B \mathbf{x}.$5 and $A\mathbf{x} = \lambda B \mathbf{x}.$6. Using $A\mathbf{x} = \lambda B \mathbf{x}.$7 Krylov vectors and initial-state blockings $A\mathbf{x} = \lambda B \mathbf{x}.$8, the KTR protocol attains relative errors in the ground-state energy below $A\mathbf{x} = \lambda B \mathbf{x}.$9 at small $T$0. Circuit depth is halved, and no ancillas are used.
• $T$1 Gauge-Higgs Model: For $T$2 qubits, with $T$3, and $T$4. With $T$5 Krylov vectors and initial blocking $T$6, relative errors $T$7 in the gauge-sector ground energy are observed. Spectral estimates align with full-KQD and DMRG benchmarks, while circuit and measurement resource requirements are greatly reduced (Mariella et al., 30 Jul 2025).

6. Protocol Constraints, Advantages, and Prospective Extensions

Advantages:

• Eliminates controlled unitaries and ancillas.
• Circuit depth is reduced by a factor of two due to midpoint time-shift; only single-qubit rotations are added.
• All observables are Pauli-strings, favorably mapping to NISQ hardware.
• Maintains robust convergence guarantees of Krylov subspace methods.

Limitations:

• Applicability requires existence of a unitary involution $T$8 with $T$9 and a $T^2=I, \qquad T^\dagger=T, \qquad \{T,H\}=0.$0-symmetric initial state; not all models (e.g., generic $T^2=I, \qquad T^\dagger=T, \qquad \{T,H\}=0.$1-local Hamiltonians such as XYZ Heisenberg) admit such involutions.
• Performance remains sensitive to initial state and time-grid selection.
• Measurement cost scales as $T^2=I, \qquad T^\dagger=T, \qquad \{T,H\}=0.$2 if $T^2=I, \qquad T^\dagger=T, \qquad \{T,H\}=0.$3 consists of $T^2=I, \qquad T^\dagger=T, \qquad \{T,H\}=0.$4 Pauli terms; batching may ameliorate overhead in specific instances (Mariella et al., 30 Jul 2025).

Extensions:

• Integration with advanced Krylov basis selection ("super-Krylov") can reduce the required subspace dimension $T^2=I, \qquad T^\dagger=T, \qquad \{T,H\}=0.$5.
• Involutions beyond Pauli-string products can be constructed via binary-linear (XOR-SAT) analysis of the Hamiltonian terms.
• Possible generalization to antiunitary symmetries or alternative discrete symmetries (parity, charge conjugation).
• Overlap matrices may be further optimized using quadrature or derivative techniques, reducing measurement burden.

The KTR protocol, by leveraging algebraic symmetries to perform control-free Krylov diagonalization, provides one of the most near-term-compatible approaches for variational quantum spectral estimation (Mariella et al., 30 Jul 2025).