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 (in the sense that the algorithm is correct at least 2/3 of the time on all inputs).
Postselection is not considered to be a feature that a realistic computer (even a quantum one) would possess, but nevertheless postselecting machines are interesting from a theoretical perspective.
Removing either one of the two main features (quantumness, postselection) from PostBQP gives the following two complexity classes, both of which are subsets of PostBQP:
The addition of postselection seems to make quantum Turing machines much more powerful: Scott Aaronson proved PostBQP is equal to PP, a class which is believed to be relatively powerful, whereas BQP is not known even to contain the seemingly smaller class NP. Using similar techniques, Aaronson also proved that small changes to the laws of quantum computing would have significant effects. As specific examples, under either of the two following changes, the "new" version of BQP would equal PP:
Basic properties.
In order to describe some of the properties of PostBQP we fix a formal way of describing quantum postselection. Define a quantum algorithm to be a family of quantum circuits (specifically, a uniform circuit family). We designate one qubit as the "postselection qubit P" and another as the "output qubit Q". Then PostBQP is defined by postselecting upon the event that the postselection qubit is formula_4. Explicitly, a language "L" is in PostBQP if there is a quantum algorithm "A" so that after running "A" on input "x" and measuring the two qubits "P" and "Q",
One can show that allowing a single postselection step at the end of the algorithm (as described above) or allowing intermediate postselection steps during the algorithm are equivalent.
Here are three basic properties of PostBQP (which also hold for BQP via similar proofs):
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 formula_5, 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 formula_6 with (conditional) 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.
Scott Aaronson showed that the complexity classes PostBQP (postselected bounded error quantum polynomial time) and PP (probabilistic polynomial time) 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" (postselection) and "Q" (output) 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["Q" = 1|"P" = 1] ≥ 2/3 if the input "x" is in the language, and Pr["Q" = 0|"P" = 1] ≥ 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["P" = 1] ≥ 2−"n""c" 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 (e.g. Hadamard, Toffoli) has this property whenever Pr["P" = 1] > 0.
Proving PostBQP ⊆ PP.
Suppose we are given a PostBQP family of circuits to decide a language "L". We assume without loss of generality (e.g. see the inessential properties of quantum computers) 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 (inputs, ancillary, output and postselection qubits), and denote the total number of gates.
Represent the "i"th gate by its transition matrix "A""i" (a real unitary formula_7 matrix) and let the initial state be |"x"> (padded with zeroes). Then formula_8. Define "S1" (resp. "S"0) to be the set of basis states corresponding to (resp. ) and define the probabilities
The definition of PostBQP ensures that either formula_11 or formula_12 according to whether "x" is in "L" or not.
Our PP machine will compare formula_13 and formula_14. In order to do this, we expand the definition of matrix multiplication:
where the sum is taken over all lists of "G" basis vectors formula_16. Now formula_17 and formula_18 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 formula_19, since then formula_20 would imply that the acceptance probability is formula_21, while formula_22 would imply that the acceptance probability is formula_23.
Technicality: we may assume entries of the transition matrices are rationals with denominator for some polynomial "f"("n")..
The definition of PostBQP tells us that formula_24 if "x" is in "L", and that otherwise formula_25. Let us replace all entries of "A" by the nearest fraction with denominator formula_26 for a large polynomial that we presently describe. What will be used later is that the "new" values satisfy formula_27 if "x" is in "L", and formula_28 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 formula_29 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 formula_30 and define the shorthand notation
then 
We define our PP machine to
Then it is straightforward to compute that this machine accepts with probability
formula_44
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(n)" 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" (implemented, e.g. by a classical circuit) which takes two inputs ("x, r") 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 (accept) or 0 (reject). The definition of PP tells us that 
Thus, 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 formula_47 is the number of rejected strings.
It is straightforward to argue that without loss of generality, formula_48; 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 formula_49. Second, we apply Hadamard gates to the first register (each of the first "T" qubits), measure the first register and postselect on it being the all-zero string. It is easy to verify that this leaves the last register (the last qubit) in the residual state
Where "H" denotes the Hadamard gate, we compute the state
Where formula_52 are positive real numbers to be chosen later with formula_53, we compute the state formula_54 and measure the second qubit, postselecting on its value being equal to 1, which leaves the first qubit in a residual state depending on formula_55 which we denote
Visualizing the possible states of a qubit as a circle, we see that if formula_57, (i.e. if formula_20) then formula_59 lies in the open quadrant formula_60 while if formula_61, (i.e. if formula_22) then formula_59 lies in the open quadrant formula_64. In fact for any fixed "x" (and its corresponding "s"), as we vary the ratio formula_55 in formula_66, note that the image of formula_67 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 formula_68, which is the center of formula_69, and let formula_70 be orthogonal to formula_71. Any qubit in formula_72, when measured in the basis formula_73, gives the value formula_71 less than 1/2 of the time. On the other hand, if formula_22 and we had picked formula_76 then measuring formula_77 in the basis formula_73 would give the value formula_71 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 (polynomially many) different values for formula_55 in hopes of getting one that is "near" "r*".
Specifically, note formula_81 and let us successively set formula_55 to every value of the form formula_83 for formula_84. Then elementary calculations show that for one of these values of "i", the probability that the measurement of formula_85 in the basis formula_73 yields formula_71 is at least formula_88
Overall, the PostBQP algorithm is as follows. Let "k" be any constant strictly between 1/2 and formula_89.
We do the following experiment for each formula_84: construct and measure formula_85 in the basis formula_73 a total of formula_93 times where "C" is a constant. If the proportion of formula_71 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 formula_85 has postselection probability formula_96 and so the overall probability is formula_97.