Boundary Complexity Inputs: Tasks, Reference States, and Gate Sets

A boundary complexity is undefined until one fixes the task, reference object, allowed gates or controls, cost functional, regulator, and approximation tolerance. State preparation, unitary synthesis, channel simulation, operator growth, purification, and Euclidean path-integral preparation are inequivalent optimizations. A bulk volume or action cannot select these inputs by itself, so every holographic comparison begins with this contract.

Required background. What Task Does Complexity Answer? owns the abstract task distinction. Regulator Dependence and Continuum Complexity supplies the continuum cutoff.

Helpful background. State, Unitary, Channel, and Operator Complexity, Complexity with Symmetry, Gauge, and Locality Constraints, Circuit Complexity in Quantum Field Theory, Cost Geometry, Gate Sets, and Reference States, and Gaussian-State Complexity provide concrete definitions.

The boundary input contract

For a regulated Hilbert space, a state-preparation complexity can be written

$$\mathcal C_{F,\mathcal G,\epsilon} (|\psi\rangle;|\psi_{\mathrm R}\rangle) =\inf_{U(s)} \left\{ \int_0^1\!\mathrm ds\,F(Y(s)): 1-|\langle\psi|U(1)|\psi_{\mathrm R}\rangle|^2\leq\epsilon \right\},$$

where

$$\frac{\mathrm dU}{\mathrm ds}U^{-1} =-i\sum_IY^I(s)G_I, \qquad G_I\in\mathcal G.$$

The gate generators $\mathcal G$, penalties inside $F$, and tolerance $\epsilon$ are defining data. A unitary-synthesis task instead minimizes distance to a target $U$ in an operator norm; an operator-growth diagnostic minimizes or measures something else again. Nielsen’s cost geometry formalizes this dependence Nielsen et al. 2006.

In QFT one must also give a lattice spacing $a$, momentum cutoff $\Lambda$, spatial volume, boundary conditions, and the order of $a\to0$. Complexity commonly diverges as a power of the number of regulated modes. Counterterms or subtractions can define comparisons, but a finite remainder is scheme dependent unless a universality argument is supplied.

A one-mode calculation

Take one harmonic oscillator with reference frequency $\omega_0$ and target frequency $\omega$. The dilation generator

$$D=\frac12(xp+px)$$

implements $x\mapsto e^r x$ and $p\mapsto e^{-r}p$. Matching the Gaussian widths requires

$$r=\frac12\log\frac{\omega}{\omega_0}.$$

With $D$ as the only allowed generator and quadratic cost $F_2=|\dot r|$, the geodesic is $r(s)=sr$ and

$$\mathcal C_2=\frac12\left|\log\frac{\omega}{\omega_0}\right|.$$

Changing $\omega_0$ changes the answer. Excluding $D$ from the gate set makes the exact task impossible; adding displacement or entangling gates changes the geometry. Multi-mode Gaussian QFT calculations retain precisely these choices Jefferson and Myers 2017.

First application

Define preparation of a regulated thermofield-double state for $n$ oscillator modes. Record: target state complexity rather than unitary complexity; product vacuum reference with frequencies $\omega_{0,k}$; Gaussian two-mode squeezing gates; an $F_2$ cost with declared penalties; lattice spacing $a$; and infidelity tolerance $\epsilon$. The modewise squeeze parameters $r_k(\beta)$ then give

$$\mathcal C_{\mathrm{TFD}}^2 =\sum_{k=1}^{n}p_k^2r_k(\beta)^2$$

for penalties $p_k$. This is a defined boundary calculation against which a bulk proposal can be compared.

Adversarial control

Change the reference frequencies, forbid nonlocal squeezing gates, or use an $F_1$ rather than $F_2$ cost. The numerical value and sometimes the scaling change while the target state is unchanged. Send $a\to0$ before specifying a subtraction and the complexity diverges. Any proposed bulk equality that lacks a transformation rule for these changes has not identified this boundary task.

Regime, evidence ceiling, and handoff

The input contract exists independently of large $N$. A holographic application additionally declares $N$, coupling, volume, temperature, cutoff, and tolerance. The bulk dual may be evaluated at small $G_N$, small curvature in string units, and within a KK truncation, but those conditions do not fix a boundary gate set.

The evidence ceiling here is a well-defined boundary target, not a bulk dictionary. Continue to Complexity Equals Volume Proposals and Complexity Equals Action Proposals to evaluate two inequivalent conjectures against it.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Jefferson, R. A., and Myers, R. C. (2017), “Circuit Complexity in Quantum Field Theory,” Journal of High Energy Physics 2017(10), 107. DOI; arXiv:1707.08570.
• Nielsen, M. A., Dowling, M. R., Gu, M., and Doherty, A. C. (2006), “Quantum Computation as Geometry,” Science 311, 1133–1135. DOI; arXiv:quant-ph/0502070.