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We define a new query measure we call quantum distinguishing complexity,denoted QD(f) for a Boolean function f. Unlike a quantum query algorithm, whichmust output a state close to |0> on a 0-input and a state close to |1> on a1-input, a "quantum distinguishing algorithm" can output any state, as long asthe output states for any 0-input and 1-input are distinguishable.  Using this measure, we establish a new relationship in query complexity: Forall total functions f, Q_0(f)=O~(Q(f)^5), where Q_0(f) and Q(f) denote thezero-error and bounded-error quantum query complexity of f respectively,improving on the previously known sixth power relationship.  We also define a query measure based on quantum statistical zero-knowledgeproofs, QSZK(f), which is at most Q(f). We show that QD(f) in fact lower boundsQSZK(f) and not just Q(f). QD(f) also upper bounds the (positive-weights)adversary bound, which yields the following relationships for all f: Q(f) >=QSZK(f) >= QS(f) = Omega(Adv(f)). This sheds some light on why the adversarybound proves suboptimal bounds for problems like Collision and Set Equality,which have low QSZK complexity.  Lastly, we show implications for lifting theorems in communicationcomplexity. We show that a general lifting theorem for either zero-errorquantum query complexity or for QSZK would imply a general lifting theorem forbounded-error quantum query complexity.

Quantum distinguishing complexity, zero-error algorithms, and statistical zero knowledge