We study how complexity classes above BQP, such as postBQP, postBQPFP, and SBQP, change if we “Merlinize” them, i.e., if we allow an extra input quantum state (or classical bit string) given by Merlin as witness. Our main results are the following: First, the Merlinized version of postBQP is equal to PSPACE. Second, if the Merlinized postBQP is restricted in such a way that the postselection probability is equal for all witness states, then the class is equal to PP. Finally, the Merlinization does not change the class SBQP.
We consider restricted versions of
postBQP where postselection probabilities can be efficiently calculated by classical or quantum computers. We show that such restricted
postBQP classes are in
AWPP(⊆PP=postBQP). This result suggests that postselecting an event with an exponentially small probability does not necessarily boost
BQP to PP. The best upperbound of
BQP is
AWPP, and therefore restricted
postBQP classes are other examples of complexity classes slightly above
BQP. In [C. M. Lee and J. Barrett, arXiv:1412.8671], it was shown that the computational capacity of a general probabilistic theory which satisfies the tomographic locality is in
AWPP. Our result therefore implies that quantum physics with restricted postselections is an example of a super quantum theory which seems to be outside of their general probabilistic theory but its computational capacity is also in
AWPP. Although it is physically natural to expect that such restricted
postBQP classes are not equivalent to BQP, we show that
UP∩coUP, which is unlikely to be in BQP, is contained in the restricted
postBQP classes. Finally, we also consider another restricted class of
postBQP, where the postselected probability depends only on the size of inputs. We show that this restricted version is contained in
APP, where
AWPP⊆APP⊆PP=postBQP.
We construct a quantum oracle relative to which BQP = QMA but cryptographic pseudorandom quantum states and pseudorandom unitary transformations exist, a counterintuitive result in light of the fact that pseudorandom states can be "broken" by quantum Merlin-Arthur adversaries. We explain how this nuance arises as the result of a distinction between algorithms that operate on quantum and classical inputs. On the other hand, we show that some computational complexity assumption is needed to construct pseudorandom states, by proving that pseudorandom states do not exist if BQP = PP. We discuss implications of these results for cryptography, complexity theory, and quantum tomography.
In 2004, Aaronson introduced the complexity class
PostBQP (
BQP with postselection) and showed that it is equal to
PP. Following their line of work, we introduce two new complexity classes. The first,
CorrBQP, is a modification of
BQP which has the power to perform correlated measurements, i.e. measurements that output the same value across a partition of registers. The second,
MajBQP, augments
BQP with the ability to collapse a register to its most likely measurement outcome. Specifically, we consider two variants,
MajBQP and
AdMajBQP, where the latter may perform intermediate measurements. We exactly characterize the computational power of the models,
CorrBQP=AdMajBQP=BPPPP and
MajBQP=PPP. In fact, we show that other metaphysical modifications of
BQP, such as
CBQP (i.e.
BQP with the ability to clone arbitrary quantum states), are also equal to
BPPPP. We show that
CorrBQP and
MajBQP are self-low with respect to classically-accessible queries. In contrast, if they were self-low under quantumly-accessible queries, the counting hierarchy would collapse. Furthermore, we introduce a variant of rational degree that lower-bounds the query complexity of
BPPPP. Lastly, we extend the adversary lower-bounding technique to
AdPDQP,
BQP with the ability to sample the current state of an algorithm with collapsing it and adapt the computation based on the samples.
We show that the class QMA does not change even if we restrict Arthur’s computing ability to only Clifford gate operations (plus classical XOR gate). The idea is to use the fact that the preparation of certain single-qubit states, so called magic states, plus any Clifford gate operations are universal for quantum computing. If Merlin is honest, he sends the witness plus magic states to Arthur. If Merlin is malicious, he might send other states to Arthur, but Arthur can verify the correctness of magic states by himself. We also generalize the result to QIP(3): we show that the class QIP(3) does not change even if the computational power of the verifier is restricted to only Clifford gate operations (plus classical XOR gate).
FINDING: Diagonalization proves uncountability and uncomputability; complexity classes form a hierarchy bounded by information-theoretic limits, with quantum classes (BQP) shown low for PP. | MATH: Cantor's theorem: |ℝ| > |ℕ| via diagonal argument; BQP^PP = PP (lowness); existence of uncomputable functions via diagonalization; complexity hierarchy: P ⊆ NP ⊆ PSPACE ⊆ EXP, with strict separations unproven except via relativization. | CONNECTION: The diagonal argument is structurally a **self-referential symmetry-breaking** — analogous to the **golden ratio's continued fraction** [1;1,1,...] which is the unique fixed point of x = 1 + 1/x (φ = 1.618). Diagonalization creates a "new element" off the diagonal, mirroring how φ² = φ + 1 generates new structure from self-reference. The **lattice of complexity classes** (P, NP, PSPACE) forms a **poset with automorphism group** — reminiscent of **root system A_n** (n-dimensional lattice symmetries). The **base-2 diagonal** (binary expansion) cont Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com