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BQP is the class of promise problems decided by uniform polynomial-size quantum circuits with bounded error. Completeness $2/3$ and soundness $1/3$ are conventional; BQP error reduction shows that other fixed separated thresholds give the same class.

Given a polynomial list of fixed-locality Hermitian terms and thresholds $a<b$ separated by at least an inverse polynomial, distinguish whether the minimum eigenvalue of their sum is at most $a$ or at least $b$. Terms may be normalized to $0\le h_j\le I$ with polynomial rescaling. This promise problem is in QMA by local-energy measurement verification.

BQP consists of promise problems decided by a uniform family of polynomial-size quantum circuits. On input $x$, with polynomially many zero-initialized ancilla qubits, a designated output measurement accepts with probability at least $2/3$ on YES inputs and at most $1/3$ on NO inputs. Uniformity means a classical polynomial-time procedure produces the quantum circuit description from the input length, or equivalently produces the verifier quantum circuit with $x$ supplied as input.

For BQP error reduction, run independent copies with freshly initialized registers. If the completeness and soundness thresholds are any constants $a>b$, accept when the fraction of accepting trials exceeds $(a+b)/2$. A Hoeffding inequality bounds either error by $\exp [-r(a-b{)}^2/2]$, with $r$ repetitions. The threshold need not be a simple majority when both $a$ and $b$ lie on the same side of $1/2$.

Choosing $r$ sufficiently large gives the usual $2/3,1/3$ thresholds, or any other fixed separated thresholds. Even an inverse-polynomial gap can be amplified using polynomially many repetitions. Classical threshold evaluation can be incorporated into the uniform computation. Thus fixed separated acceptance probabilities define the same BQP class. This argument concerns unrestricted BQP quantum circuits and does not assume such amplification is available for restricted stoquastic circuits.

QMA is the class of promise problems for which a uniform polynomial-size quantum circuit checks a polynomial-size quantum witness. A YES instance has some quantum witness accepted with probability at least $2/3$; on a NO instance every quantum witness is accepted with probability at most $1/3$. The verifier initializes its own ancilla qubits to zero. Mixed quantum witnesses do not improve the maximum because acceptance is linear in their density operator.

For a verifier $U_x$ and quantum witness embedding $J|w⟩=|w⟩|0\cdots 0⟩$, define the witness acceptance operatorThe Rayleigh quotient gives ${\max }_{\Vert w\Vert =1}⟨w|F|w⟩={\lambda }_{\max }(F)$. Every entry of $F$ can be recomputed by the polynomial-storage path summation of part (b); no full quantum witness matrix needs to be stored.

Let the quantum witness have $m$ qubits, so its dimension is $D=2^m$. For this positive operator, trace-power witness optimization usesThis is the useful exponentiated form of the supplied logarithmic inequality. Positivity is required; such a logarithmic statement is not true for an arbitrary operator. No logarithms or roots need to be calculated.

Choose $d=2m+2$. On a NO instance,since $2^m(2/3{)}^{2m+2}=(4/9)(8/9{)}^m<1$. On a YES instance, $\mathrm{Tr}(F^d)\ge (2/3{)}^d>(1/2{)}^d$. Thus comparison of $2^d\mathrm{Tr}(F^d)$ with one distinguishes the cases using only multiplication and a final comparison.

Evaluate the trace byThere are $d=O(m)$ active quantum witness indices, each requiring $m$ bits. Accumulate one product at a time and recompute its matrix entries as required. Together with the verifier path workspace, this uses polynomially many real registers. The potentially enormous time and the numerical precision are unrestricted in $C$. Therefore $\mathrm{QMA}\subseteq C$, without solving an exponentially large eigenvalue problem by storing its matrix.

A promise problem $(L_{\mathrm{yes}},L_{\mathrm{no}})$ belongs to QMA if there are polynomials $p,q$ and a uniform family of quantum circuits $V_x$ of size at most $q(|x|)$, taking a $p(|x|)$-qubit witness and clean ancillas, with one output qubit interpreted as acceptance, such thatNo condition is imposed outside the promise. The witness may be an arbitrary density operator, and the verifier is efficient but need not know how to prepare an honest witness. Pure witnesses suffice for completeness because the acceptance probability is linear in the witness. QMA is quantum verification with polynomially many witness qubits and a constant completeness-soundness gap.

The local Hamiltonian problem is a promise problem specified by a sum $H={\sum }_{j=1}^M{h}_j$ on $n$ qubits, where $M$ is polynomial in $n$, each Hermitian term acts on at most a fixed number $k$ of qubits, and the terms and two thresholds $a<b$ are described with polynomially many bits. The promise gap satisfies $b-a\ge 1/\mathrm{poly}(n)$. In a convenient normalization, $0\le h_j\le I$. The task is to distinguishThere is no required answer when the minimum energy lies between the thresholds. A formulation with Hermitian terms of polynomially bounded norm is equivalent: shift each term by a known scalar lower bound and rescale all terms by a polynomial bound to obtain positive semidefinite terms of norm at most one, while shifting and rescaling $a,b$ in parallel. The promise gap remains inverse-polynomial.

A promise problem consists of disjoint sets of YES and NO instances. An algorithm or verifier must obey its correctness guarantees only on their union. Energy problems with separated thresholds are naturally promise problems because intermediate-energy instances require no prescribed answer.

QMA is the class of promise problems with a polynomial-size quantum witness checked by a uniform polynomial-size quantum circuit. YES instances have an accepting witness with probability at least $2/3$; NO instances have acceptance at most $1/3$ for every witness, including mixed states. The verifier's acceptance is linear in the witness density operator. Inverse-polynomial or exponentially small error can be obtained through QMA error reduction.