On the Simulation of Adaptive Measurements via Postselection

arXiv:1107.3271v1 [cs.CC] 17 Jul 2011 ON THE SIMULATION OF ADAPTIVE MEASUREMENTS VIA POSTSELECTION VIKRAM DHILLON Abstract. In this note we address the question of whether any any quantum computational model that allows adaptive measurements can be simulated by a model that allows postselected measurements. We argue in the favor of this question and prove that adaptive measurements can be simulated by postselection. We also discuss some potentially stunning consequences of this result such as the ability to solve #P problems. 1. Introduction In [3] Aaronson introduced a complexity class PostBQP, which is is a complexity class consisting of all of the computational problems solvable in polynomial time on a quantum Turing machine with postselection and bounded error. It was also shown equivalent to PP which is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. Aaronson then raised an interesting question which asks whether adaptive measurements made by a quantum computational model be simulated with postselected measurements. In this note we address this question by asserting that it is possible to simulate adaptive measurements by postselection on the quantum circuit model of computation. We also explore the consequences of being able to atleast theoritically perform this simulation, it is known that PPP = P#P which implies that the complexity of PP is equivalent to that of P#P which is an NP.Sso if an adaptive (non-projective) measurement such as a weak measurement can be simulated, following the work of Lloyd et.al logical gates can be constructed that allow us to solve P#P problems. 2. Proof Before we show how the simulation would work, we want to establish some definitions to make an easier transition to the proof itself. Definition 1. An adaptive measurement is an incomplete measurement is made on the system, and its result used to choose the nature of the second measurement made on the system, and so on (until the measurement is complete). A complete measurement is one which leaves the system in a state independent of its initial state, and hence containing no further information of use. [4] Definition 2. Postselection is the power of discarding all runs of a computation in which a given event does not occur. [3] We will be using the quantum circuit model which is the standard model in quantum computation theory and most other computational models have been shown to be equivalent to it. The equivalence also allows us to simulate those models on the circuit model. Lemma 3. Measurement based quantum computation (MBQC) employs adaptive local measurements on a resource state. Proof. See [4] for this. □ Lemma 4. Measurement based models can be simulated on the quantum circuit model. 1 2 VIKRAM DHILLON Proof. Any one-way computation can be made into a quantum circuit by using quantum gates to prepare the resource state [5]. □ Axiom 5. From Lemma 3 and Lemma 4 we can deduce that the qunatum circuit model can simulate mea- surement based computation which is a model that allows for adaptive measurements. The above mentioned axiom completes the first part of the corrospondence, we now have to show that the same model that can simulate postselected measurements to complete the corrospondance. Postselected measurements fall under the complexity class PostBQP and we will also use the equivalence of PostBQP and PP shown by Aaronson in [3]. This switch between complexity classes makes this proof simplistic. Axiom 6. BQP ⊂PP Lemma 7. PP ∩BQP /∈{φ} Proof. Let us assume that no problem exist at the intersection of PP and BQP However, we know the aforementioned axiom to be true so there must atleast be one problem that exist at the intersection of the complexity classes. That particular problem, by the virtue of being at the intersection will be both PP and BQP which is self-evident. We will represent the problems present at the intersection of the two complexity classes by the set τ. □ Lemma 8. From Lemma 7 we can deduce that elements of τ can be simulated on a quantum computer Proof. The elements of τ fall in the clas BQP which can be simulated on a quantum computer therefore the elements of that set can also be simulated by a quantum computation model. □ Lemma 9. Elements of τ can exibit postselected measurements Proof. The members of τ are both PP and BQP where BQP can be simulated on a quantum computer and since PP = PostBQP, elements of τ can be simulated through the use of postselected measurements. □ Axiom 10. From the preceding proof and Axiom 5, we see the corrospondence is complete. We can indeed take a quantum computational model (in our case, it is the standard quantum circuit model) that allows for adaptive measurements (Axiom 5) and simulate it with a model that allows for postselected measurements (again the quantum circuit model) The preceeding axiom presents the completed proof, in the following secti

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