The class of decision problems solvable in polynomial time, with bounded error, by a non-Hermitian quantum computer (NQC): a uniform family of polynomial-size quantum circuits over ordinary qubits whose universal gate set {H, T, CNOT, G} relaxes the unitarity constraint by including the nonunitary linear gate G = diag(g−1, g), where g > 0 and g ≠ 1 is a real constant. Qubit states need not be normalized; the probability of a measurement outcome is the squared amplitude divided by the squared norm of the state.
Defined in [ZW26], where it is shown that an NQC solves the NP-hard maximum independent set problem in polynomial time, so that NP ⊆ BNQP, and furthermore that BNQP = P#P. Hence BNQP contains PH (by Toda's theorem [Tod89]) as well as PP and PostBQP, and is contained in PSPACE.
The nonunitary gate G can in principle be implemented in real physical systems, e.g., a two-mode many-boson system with particle gain and loss, or by postselection; however, both schemes analyzed in [ZW26] require an exponentially large amount of physical resources, suggesting that the extraordinary power of the model originates from this exponential physical overhead. Relatedly, [BL25] show that PostBQP = PP captures the power of generic nonunitary gates polynomially far from unitarity.
Equals P#P.
The class of decision problems solvable in polynomial time, with bounded error, by a Lorentz quantum computer (LQC): a uniform family of polynomial-size circuits over qubits and hyperbolic bits (hybits) — bits whose two states evolve via complex Lorentz transformations rather than unitary ones, with one state observable and the other unobservable. The circuits use both unitary and Lorentzian gates and allow measurements, after which no further gates are applied. If the answer is 'yes', an accepting outcome occurs with probability higher than 2/3; if the answer is 'no', with probability less than 1/3.
Defined in [ZW26b], where an LQC algorithm solves the NP-hard maximum independent set problem in polynomial time, so that NP and coNP are contained in BLQP, and where it is furthermore shown that BLQP = P#P. Hence BLQP contains PH (by Toda's theorem [Tod89]) as well as PP and PostBQP, and is contained in PSPACE.
LQC can efficiently simulate quantum computing with postselection, while the reverse is not possible: this "super-postselection" capability separates BLQP from PostBQP [ZW26b].
Equals P#P. See also BNQP (the non-Hermitian quantum computing characterization of the same class).