PostBQP
In computational complexity theory, PostBQP 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.
Postselection is not considered to be a feature that a realistic computer would possess, but nevertheless postselecting machines are interesting from a theoretical perspective.
Removing either one of the two main features from PostBQP gives the following two complexity classes, both of which are subsets of PostBQP:
- BQP is the same as PostBQP except without postselection
- BPPpath is the same as PostBQP except that instead of quantum, the algorithm is a classical randomized algorithm
- if we broadened the definition of 'quantum gate' to include not just unitary operations but linear operations, or
- if the probability of measuring a basis state was proportional to instead of for any even integer p > 2.
Basic properties
- P = 1 with nonzero probability
- if the input x is in L then Pr ≥ 2/3
- if the input x is not in L then Pr ≥ 2/3.
Here are three basic properties of PostBQP :
1. PostBQP is closed under complement. Given a language L in PostBQP and a corresponding deciding circuit family, create a new circuit family by flipping the output qubit after measurement, then the new circuit family proves the complement of L is in PostBQP.
2. You can do probability amplification in PostBQP. The definition of PostBQP is not changed if we replace the 2/3 value in its definition by any other constant strictly between 1/2 and 1. As an example, given a PostBQP algorithm A with success probability 2/3, we can construct another algorithm which runs three independent copies of A, outputs a postselection bit equal to the conjunction of the three "inner" ones, and outputs an output bit equal to the majority of the three "inner" ones; the new algorithm will be correct with conditional probability, greater than the original 2/3.
3. PostBQP is closed under intersection. Suppose we have PostBQP circuit families for two languages L1 and L2, with respective postselection qubits and output qubits P1, P2, Q1, Q2. We may assume by probability amplification that both circuit families have success probability at least 5/6. Then we create a composite algorithm where the circuits for L1 and L2 are run independently, and we set P to the conjunction of P1 and P2, and Q to the conjunction of Q1 and Q2. It is not hard to see by a union bound that this composite algorithm correctly decides membership in with probability at least 2/3.
More generally, combinations of these ideas show that PostBQP is closed under union and BQP truth-table reductions.
PostBQP = PP
showed that the complexity classes PostBQP and PP are equal. The result was significant because this quantum computation reformulation of PP gave new insight and simpler proofs of properties of PP.The usual definition of a PostBQP circuit family is one with two outbit qubits P and Q with a single measurement of P and Q at the end such that the probability of measuring P = 1 has nonzero probability, the conditional probability Pr ≥ 2/3 if the input x is in the language, and Pr ≥ 2/3 if the input x is not in the language. For technical reasons we tweak the definition of PostBQP as follows: we require that Pr ≥ 2−nc for some constant c depending on the circuit family. Note this choice does not affect the basic properties of PostBQP, and also it can be shown that any computation consisting of typical gates has this property whenever Pr > 0.
Proving PostBQP ⊆ PP
Suppose we are given a PostBQP family of circuits to decide a language L. We assume without loss of generality that all gates have transition matrices that are represented with real numbers, at the expense of adding one more qubit.Let denote the final quantum state of the circuit before the postselecting measurement is made. The overall goal of the proof is to construct a PP algorithm to decide L. More specifically it suffices to have L correctly compare the squared amplitude of in the states with to the squared amplitude of in the states with to determine which is bigger. The key insight is that the comparison of these amplitudes can be transformed into comparing the acceptance probability of a PP machine with 1/2.
Matrix view of PostBQP algorithms
Let n denote the input size, denote the total number of qubits in the circuit, and denote the total number of gates.Represent the ith gate by its transition matrix Ai and let the initial state be |x>. Then. Define S1 to be the set of basis states corresponding to and define the probabilities
The definition of PostBQP ensures that either or according to whether x is in L or not.
Our PP machine will compare and. In order to do this, we expand the definition of matrix multiplication:
where the sum is taken over all lists of G basis vectors. Now and can be expressed as a sum of pairwise products of these terms. Intuitively, we want to design a machine whose acceptance probability is something like, since then would imply that the acceptance probability is, while would imply that the acceptance probability is.
Technicality: we may assume entries of the transition matrices are rationals with denominator {{math|2}} for some polynomial ''f''(''n'').
The definition of PostBQP tells us that if x is in L, and that otherwise. Let us replace all entries of A by the nearest fraction with denominator for a large polynomial that we presently describe. What will be used later is that the new values satisfy if x is in L, and if x is not in L. Using the earlier technical assumption and by analyzing how the 1-norm of the computational state changes, this is seen to be satisfied if thus clearly there is a large enough f that is polynomial in n.Constructing the PP machine
Now we provide the detailed implementation of our PP machine. Let denote the sequence and define the shorthand notationthen
We define our PP machine to
- pick a basis state uniformly at random
- if then STOP and accept with probability 1/2, reject with probability 1/2
- pick two sequences of G basis states uniformly at random
- compute
- if then accept with probability and reject with probability
- otherwise accept with probability and reject with probability
so this is a PP machine for the language L, as needed.
Proving PP ⊆ PostBQP
Suppose we have a PP machine with time complexity T:=T on input x of length n := |x|. Thus the machine flips a coin at most T times during the computation. We can thus view the machine as a deterministic function f which takes two inputs where r, a binary string of length T, represents the results of the random coin flips that are performed by the computation, and the output of f is 1 or 0. The definition of PP tells us thatThus, we want a PostBQP algorithm that can determine whether the above statement is true.
Define s to be the number of random strings which lead to acceptance,
and so is the number of rejected strings.
It is straightforward to argue that without loss of generality, ; for details, see a similar without loss of generality assumption in the proof that PP is closed under complementation.
Aaronson's algorithm
Aaronson's algorithm for solving this problem is as follows. For simplicity, we will write all quantum states as unnormalized. First, we prepare the state. Second, we apply Hadamard gates to the first register, measure the first register and postselect on it being the all-zero string. It is easy to verify that this leaves the last register in the residual stateWhere H denotes the Hadamard gate, we compute the state
Where are positive real numbers to be chosen later with, we compute the state and measure the second qubit, postselecting on its value being equal to 1, which leaves the first qubit in a residual state depending on which we denote
Visualizing the possible states of a qubit as a circle, we see that if, then lies in the open quadrant while if, then lies in the open quadrant. In fact for any fixed x, as we vary the ratio in, note that the image of is precisely the corresponding open quadrant. In the rest of the proof, we make precise the idea that we can distinguish between these two quadrants.
Analysis
Let, which is the center of, and let be orthogonal to. Any qubit in, when measured in the basis, gives the value less than 1/2 of the time. On the other hand, if and we had picked then measuring in the basis would give the value all of the time. Since we don't know s we also don't know the precise value of r*, but we can try several different values for in hopes of getting one that is "near" r*.Specifically, note and let us successively set to every value of the form for. Then elementary calculations show that for one of these values of i, the probability that the measurement of in the basis yields is at least
Overall, the PostBQP algorithm is as follows. Let k be any constant strictly between 1/2 and.
We do the following experiment for each : construct and measure in the basis a total of times where C is a constant. If the proportion of measurements is greater than k, then reject. If we don't reject for any i, accept. Chernoff bounds then show that for a sufficiently large universal constant C, we correctly classify x with probability at least 2/3.
Note that this algorithm satisfies the technical assumption that the overall postselection probability is not too small: each individual measurement of has postselection probability and so the overall probability is.
Implications
- See Quantum computation reformulation of PP