Translationally Invariant Compressed Control (TICC)

• TICC is a specialized protocol for quantum simulation that reduces ancilla control overhead from a multiplicative factor to an additive term, achieving near-optimal circuit depth scaling.
• It leverages translational invariance and a TI brick-wall ansatz optimized on small systems to efficiently scale to larger, local Hamiltonians.
• Demonstrated in iterative quantum phase estimation, TICC enables sub-percent error simulations with significantly reduced gate counts on current hardware models.

Translationally Invariant Compressed Control (TICC) is a specialized compression protocol for quantum simulation algorithms, principally addressing the efficient implementation of controlled time evolution for translationally invariant, local Hamiltonians. TICC optimizes the ancilla-controlled implementation of quantum circuits required for algorithms such as Quantum Phase Estimation (QPE). Its central contribution is the reduction of the control overhead in quantum circuits from a multiplicative to an additive factor, achieving near-optimal circuit depth scaling in both simulation time and system size, while retaining rigorous fidelity guarantees. The protocol enables practical realization of large-scale quantum simulations and phase estimation tasks on near-term hardware (Karacan, 26 Nov 2025).

1. Protocol Definition and Theoretical Foundation

TICC targets the challenge arising when quantum simulation algorithms, notably QPE, demand ancilla-controlled implementations of the time evolution operator. The canonical controlled evolution is $$U_c(t) = |0\rangle\langle0|_{\rm anc}\otimes I + |1\rangle\langle1|_{\rm anc}\otimes e^{-iHt}$$, for Hamiltonian $H$ and time $t$. In standard approaches, every gate in the simulation of $e^{-iHt}$ is promoted to its controlled version, incurring a significant multiplicative control overhead denoted $\gamma_D \sim 15$–$18$.

TICC exploits a key equivalence: for algorithms utilizing a single ancilla, it suffices to implement $$|0\rangle\langle0|\otimes e^{+iH\,t/2} + |1\rangle\langle1|\otimes e^{-iH\,t/2}$$, which differs from $U_c(t)$ only by an overall energy-space phase. Crucially, controlling the “direction” rather than “duration” of evolution enables the reduction of overhead from multiplicative to additive with respect to the number of controlled layers.

Translational invariance is fundamental to TICC. The protocol assumes $H$ is both local and translationally invariant (TI) on an $N$-site lattice. A TI brick-wall ansatz $W(V) = \cdots V_2 V_1 V_0$ is optimized on a small system ($N=4$–$6$), such that $\|W(V^*)-e^{-iHt}\| \leq \epsilon$, for target error $\epsilon$. By the Lieb–Robinson bound, the optimized gates transfer directly to larger $N$ when $t$ does not exceed $t_{\max} = \mathcal{O}(N^{1/D}/v_{\rm LR})$, where $v_{\rm LR}$ is the Lieb–Robinson velocity.

2. Quantum Circuit Construction and Optimization

TICC distinguishes between circuit layers for control promotion. The layers of the TI ansatz $W(V)$ are partitioned into:

• Control layers ($V \setminus \tilde V$, labeled “green”), subject to ancilla control.
• Uncontrolled layers ($\tilde V$, labeled “red”), which remain unconditional.

The circuit construction proceeds as follows:

a) Hamiltonian decomposition: $H = \sum_{i=1}^\eta H_i$, with $\eta$ anti-commuting sub-Hamiltonians (e.g., subsets of Pauli-term strings), each related by $K_i^\dagger H_i K_i = -H_i$.

b) Brickwall ansatz: Depth $\gamma$ per “block,” interleaved with $\eta + 1$ control layers based on the $K_i$ operators. The parameter set is $V = \{V_0,\dots,V_{\eta\gamma+\eta}\}$ with $\tilde V \subset V$.

c) Cost function optimization:

$$f(V) = -\mathrm{Re}\,\mathrm{Tr}\!\bigl[U(t/2)^\dagger W(V)\bigr] -\mathrm{Re}\,\mathrm{Tr}\!\bigl[U(-t/2)^\dagger W(\tilde V)\bigr]$$

such that $W(V^*) \approx e^{-iH t/2}$ and $W(\tilde V^*) \approx e^{+iH t/2}$.

d) Transfer to large systems: After optimization on a small TI system, only the designated $\eta + 1$ layers $V^*\setminus \tilde V^*$ are promoted to controlled two-qubit unitaries when constructing circuits on larger devices.

3. Circuit Depth Scaling and Control Overhead

TICC achieves near-optimal scaling for circuit depth, matching rigorous product-formula lower bounds. In gate-oracle models, any algorithm requires $\Omega(t\,\mathrm{polylog}(1/\epsilon))$ queries to $H$ given target error $\epsilon$. The Riemannian QC-opt existence theorem guarantees that for TI, 2-local $H$ on $N$ sites, a TI brickwall circuit exists of depth $\mathcal{O}(t\,\mathrm{polylog}(N t/\epsilon))$.

TICC preserves this scaling for the unconditional (red) layers while drastically reducing the cost of control. If each controlled two-qubit gate decomposes into $\gamma_D$ native gates, and the number of control layers is $\eta + 1$, independent of $t$, the total circuit depth is:

$$\mathcal{O}(t\,\mathrm{polylog}(N t/\epsilon)) \text{ (uncontrolled layers) } + \gamma_D (\eta + 1)$$

Whereas naïve control multiplies total depth (and gate count) by $\gamma_D$, TICC incurs only $\gamma_D (\eta + 1)$ as an additive term, yielding significant savings especially in regimes where $\gamma \gg \eta$.

4. Analytical Framework and Expressions

TICC’s mathematical targets and scaling laws include:

• Spectral-norm error: $\|W - e^{-iHt}\| \leq \epsilon$.
• Maximal time transfer: $t_{\max} = \mathcal{O}(N^{1/D}/v_{\rm LR})$.
• Query/gate complexity lower bound: $\Omega(t\,\mathrm{polylog}(1/\epsilon))$.
• TICC circuit depth scaling: $\mathcal{O}(t\,\mathrm{polylog}(t N/\epsilon))$.
• Total two-qubit gate count:

$$\frac{N}{2} \times d \times [\gamma \eta + \gamma_D (\eta + 1)]$$

where $d$ is the number of inequivalent neighbor-permutation classes by lattice translation.

A plausible implication is that direct optimization of the brickwall ansatz enables hardware-efficient simulations on lattices with translational symmetry, supporting sub-percent simulation errors with modest resource budgets.

5. Implementation in Iterative Quantum Phase Estimation

TICC is concretely demonstrated for QPE on a frustrated transverse-field Ising model defined on a $6\times6$ triangular lattice. The protocol’s controlled time evolution implementation for such a system requires only 414 two-qubit (CNOT-equivalent) gates on the 6×6 device.

On a $4\times4$ triangular lattice, using a Quantinuum H2 trapped-ion noise-aware emulator, TICC-optimized iterative QPE circuits are constructed with 184 hardware-native ZZPhase gates per run (consistent extrapolation to 414 gates for the 6×6 case). Ground-state energy errors consistently fall below $1\%$ for all $g \in \{1.5,2,2.5,3\}$ with observed relative errors $\{0.7\%, 0.1\%, 0.2\%, 0.2\%\}$ and uncertainty $\pm 1.5\%$ after noise-aware amplitude renormalization.

The emulator’s noise model incorporates:

• One-qubit depolarizing errors $p_1 \approx 2 \times 10^{-5}$,
• Two-qubit depolarizing errors $p_2 \approx 10^{-3}$,
• Spontaneous emission probabilities $10^{-5}$–$10^{-4}$,
• Idling/transport dephasing ($f_L$, $f_Q$), initialization/readout faults, and crosstalk.

A plausible implication is that TICC’s additive control overhead makes feasible robust phase estimation with sub-percent error using present-day hardware models and noise corrections.

6. Advantages, Limitations, and Trade-offs

Advantages:

• Achieves near-optimal scaling in $t$ and $\epsilon$ as provably matched by theory.
• Control overhead is additive in circuit depth rather than multiplicative, reducing total gate count.
• Enables protocol transfer from small to large systems via the TI ansatz (optimizing on $4$–$6$ qubits, reusing on $6\times6$).
• Concretely demonstrates sub-percent error QPE on frustrated $4\times4$ lattices using existing hardware models.

Limitations:

• Requires strict translational invariance and locality; disorder or long-range couplings invalidate transferability.
• Maximum evolution time $t_{\max} = \mathcal{O}(N^{1/D})$; simulating larger $t$ must incrementally combine shorter segments, leading to linear error accumulation.
• Ansatz optimization is nonconvex, demanding suitable initialization (such as low-order Trotterization) and classical computational resources for gradient/Hessian estimation.

Trade-offs:

• Relative to product-formula approaches with Pauli-string insertion, TICC uses asymptotically fewer gates at large $t$ and high precision, necessitating up-front classical optimization.
• Compared to universal control of all compressed gates, TICC reduces control overhead from multiplicative ($\times \gamma_D$) to additive ($+ \gamma_D(\eta+1)$), offering notable gate savings in regimes with many variational layers.

In summary, TICC leverages translational symmetry, targeted Hamiltonian decomposition, and variational optimization to construct ancilla-controlled time-evolution circuits of depth $\mathcal{O}(t\,\mathrm{polylog}(tN/\epsilon))$, maintaining only an additive burden for control. This approach bridges quantum algorithmic optimality and hardware-efficient implementation, realizing sub-percent error QPE on nontrivial 2D lattices with practical gate counts (Karacan, 26 Nov 2025).