Observables inW*-Algebraic Quantum Mechanics

Anton Amann

Anton Amann

Pöttinger, Jürgen

Robert Olkiewicz

Uma Girish, Ran Raz

We show that quantum algorithms of time $T$ and space $S\ge \log T$ with unitary operations and intermediate measurements can be simulated by quantum algorithms of time $T \cdot \mathrm{poly} (S)$ and space $ {O}(S\cdot \log T)$ with unitary operations and without intermediate measurements. The best results prior to this work required either $Ω(T)$ space (by the deferred measurement principle) or $\mathrm{poly}(2^S)$ time [FR21,GRZ21]. Our result is thus a time-efficient and space-efficient simulation of algorithms with unitary operations and intermediate measurements by algorithms with unitary operations and without intermediate measurements. To prove our result, we study pseudorandom generators for quantum space-bounded algorithms. We show that (an instance of) the INW pseudorandom generator for classical space-bounded algorithms [INW94] also fools quantum space-bounded algorithms. More precisely, we show that for quantum space-bounded algorithms that have access to a read-once tape consisting of random bits, the final state of the algorithm when the random bits are drawn from the uniform distribution is nearly identical to the final state when the random bits are drawn using the INW pseudorandom generator. This result applies to general quantum algorithms which can apply unitary operations, perform intermediate measurements and reset qubits.

Emile Grgin, Aage Petersen

In classical and in quantum mechanics physical variables play a dual role as observables and as generators of infinitesimal transformations in the invariance groups. We show that if the Lie algebra of generators is central simple, the observable-generator duality restricts the structure of the algebra of observables to two cases: a commutative, associative algebra as in classical mechanics, or a central simple special Jordan algebra as in quantum mechanics.