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An avoided crossing occurs when two parameter-dependent eigenvalues approach one another but an interaction prevents equality. Near a two-level crossing, a nonzero off-diagonal matrix element opens a positive minimum gap.

Expand a quantum circuit matrix element by inserting computational basis resolutions between gates, then enumerate the intermediate strings depth first. The active stack stores one basis label and accumulator per quantum circuit layer, using polynomial space for polynomially many qubits and gates. Recompute entries rather than storing an exponential matrix or state vector. The output probability requires the Born rule sum of squared amplitude moduli.

The inclusive product of hadronic-current matrix elements, summed over final states with a Lorentz-invariant phase-space measure. Electromagnetic current conservation makes both indices transverse to the exchanged four-momentum.

The product of lepton-current matrix elements with a specified spin sum or spin average. Its normalization must be stated when quoting a differential scattering cross-section.

Approximate a hadronic matrix element of a product of two currents by a product of separate current matrix elements. For a decay to two mesons, one factor can create a pion from the vacuum while the other gives a pseudoscalar-to-pseudoscalar form factor. Nonfactorizable Quantum chromodynamics interactions are omitted; tree-level weak vertices do not alone justify this approximation.

The class $C$ uses polynomial-register real-arithmetic computation: its storage restriction counts real registers, without limiting their precision or the running time. Use the usual decision-model interpretation that computed real values may be compared with fixed thresholds, and that the fixed universal gate set's real constants are available. Without a way to test values, arithmetic instructions alone would not specify the intended decision model.

Let a BQP quantum circuit use $q=\mathrm{poly}(n)$ qubits and $T=\mathrm{poly}(n)$ gates. A matrix element can be evaluated by depth-first quantum circuit path summation:Enumerate the intermediate $q$-bit strings recursively. Store the current strings and loop positions, a partial product, and the partial sum at each depth. There are at most $T$ active levels, using $O(qT)$ registers or bits of loop information, not an exponentially long state vector. Each fixed-locality gate matrix element is computed from its few affected bits. Represent a complex number by two real registers; complex multiplication and addition require only real addition, subtraction and multiplication.

Recompute this amplitude separately for every final string $z$ whose output bit is one, accumulating the Born rule probabilityThe outer enumeration needs only another $O(q)$ bits and a real accumulator. Arbitrarily large running time is allowed, so repeated recomputation is harmless. Compare $p_1$ with $1/2$ to distinguish the promised BQP cases. This proves $\mathrm{BQP}\subseteq C$ using polynomial storage.

The printed probability hint omits the squares of the amplitude moduli. Summing their absolute values alone is not the Born rule and can exceed one. The corrected sum above is necessary for the containment proof. The argument is about the stated real-register model, rather than a claim that exponential-precision numbers are free in an ordinary bit-cost model.

For an excited eigenstate, $E_i-E_0\ge \Delta$. The matrix-element identity in condition (i), together with the two-sided Fourier cutoff, givesThe reverse matrix element has frequency $E_0-E_i\le -\Delta$ and also vanishes. Equivalently, $A^{(Z)}$ is Hermitian because $h_Z$ is Hermitian and $w$ is real, so the two elements are conjugate. ThusThe spectral gap eliminates precisely the couplings needed to make the unique ground state an eigenvector of every filtered term. Couplings between excited states with smaller energy differences can remain; the spectral filter need not diagonalize the whole operator.

Use the Fourier transform convention $\hat{w}(E)={\int }_{\mathbb{R}}w(t)e^{itE}dt$. Since $w\ge 0$, it is real, so $\hat{w}(-E)=\hat{w}(E{)}^{\ast }$. The positive-frequency cutoff therefore also eliminates frequencies at or below $-\Delta$. Normalization gives $\int w=\hat{w}(0)=1$.

For spectral filtering of Hamiltonian terms, chooseIn an energy eigenbasis, its matrix elements areAt $i=j=0$ the frequency is zero, so the normalization preserves the ground-state expectation:The spectral filter is a positive weighted average of unitary conjugations; in particular the integral is bounded in operator norm by $\Vert h_Z\Vert$. We can choose it even without an extra assumed tail bound: the evenization of a nonnegative bandlimited filter proved in part (e) produces another admissible spectral filter with the same type of positive-time decay. Make that choice consistently in all the filtered terms and shell definitions below.