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Walkthrough

Quantum states and computational basis

OQaml represent the quantum states in the standard textbook ordering of the computational basis. I.e. the index of a qubit increases from left to right as opposed to the standard binary representation:

$$ |\Psi\rangle= |b_0b_1...b_{n-1}\rangle = |b_0\rangle\otimes|b_1\rangle\otimes...\otimes|b_{n-1}\rangle $$

The individual qubit space is denoted by $|b_j\rangle \in \{|0\rangle, |1\rangle\}$. The full quantum state of a QVM with $n$ qubits can then be represented by a vector of size $2^n$ whose entries are complex numbers. The amplitude of a given computational bais state is then given by the entry in the state-vector corresponding to the integer of the bit-string representation. E.g. the state $i|10\rangle$ corresponds to the third entry in the four-dimensional wave-function vector containing the complex unit $i$. To create this state in OQaml you can run the following commands in utop

#require "oqaml, owl";;
module V = Owl.Dense.Vector.C;;
module Q = Oqaml;;

let tqvm = Q.init_qvm 2;;

The first few lines are necessary imports while the last line initializes a fresh QVM state in state $|00\rangle$ corresponding to the vector $(1, 0, 0, 0)^T$. To create the state $i|10\rangle$ we need to make a flip with a so-called $Y$-gate. This can be achieved by

Q.apply (Q.Y 0) tqvm;;

changing the internal wave-function vector to $(0, 0, i, 0)^T$.

Gates and Circuits

The application of gates $U$ represents discrete operations that bring the QVM from state $|\Psi_i\rangle$ to state $|\Psi_f\rangle$ according to the prescription

$$|\Psi_f\rangle = U |\Psi_i\rangle$$

In OQaml this transition operation is deeply engrained into the syntax as seen by the type definition of the apply function, which maps a gate to a qvm returning a qvm

val apply: gate -> qvm -> qvm

The gates $U$ are represented by matrices. Note that in general gates are not unitary as they also incorporate classical gates and projections (corresponding to measurements). A circuit is described as a composition of gates and in Kitaev ordering (time from right to left) reads as

$$|\Psi_f\rangle= U_nU_{n-1}\dots U_1 |\Psi_i\rangle$$

Note the operator ordering in the above expression corresponding to the Kitaev notation indicating that time flows from right to left. Due to the mathematical similarity we can define a generalized gate as

$$ G = U_nU_{n-1}\dots U_1 $$

which highlights the fact that circuits and gates are conceptually the same mathematical type. OQaml makes use of this fact by defining gates as a recursive type as seen in the gate type definition

type gate =
  | ...
  | CIRCUIT of gate list
  |...

This abstraction lets you define arbitraty gate as a circuit, ensuring that the application of this gate in extended circuits is a valid operation as any gate type can be used in the apply function defined above.

Example - Phase gate

The phase gate is defined as

$$ S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} $$

We can decompose any single qubit gate $U$ into a sequence of defined rotations according to

$$ U = \textrm{e}^{i\alpha} R_z(\beta)R_y(\gamma)R_z(\delta) $$

and with a bit of algebra we find that to decompose $S$ we have $\gamma = 0$ and $\alpha = \beta=\delta = \pi/4$. In OQaml we can now define a gate according to

let pi4 = Owl.Maths.pi_4;;
let pg idx = Q.CIRCUIT [Q.PHASE (pi4); Q.RZ(pi4, idx); Q.RY (0.0, idx); Q.RZ (pi4, idx)];;

>>> val pg : int -> Q.gate = <fun>

There are two things worthwhile to point out:

  1. note how the above code snippet mimicks the Kitaev ordering of the corresponding Gate/Circuit definition.
  2. the type of pg is int -> Q.gate meaning it maps an integer index to a gate, indicating the recursive nature of a gate type, as discussed above.

The newly definded gate can now be used as follows

Q.apply (Q.CIRCUIT [pg 0; Q.X 0]) (Q.init_qvm 1);;

>>> {Q.num_qubits = 1;
wf =
        C0
R0 (0, 0i)
R1 (0, 1i);

reg = [|0|]}

Example - Equivalence between SWAP and CNOT

There is a well-known identity between $\textrm{SWAP}$ and $\textrm{CNOT}$ gates:

$$ \textrm{SWAP}[i,j] = \textrm{CNOT}[i,j] \otimes \textrm{CNOT}[j,i] \otimes \textrm{CNOT}[i,j] $$

We can prove this equivalence quite easy with OQaml. First we define the $\textrm{SWAP}$ gate in terms of the $\textrm{CNOT}$ gates

let swap i j = Q.CIRCUIT [Q.CNOT (i,j); Q.CNOT (j,i); Q.CNOT (i,j)];;

we can then write a test to assert equivalence

let tqvm = Q.apply (Q.X 0) (Q.init_qvm 2);;
Q.apply (swap 0 1) tqvm = Q.apply (Q.SWAP (0,1)) tqvm;;

>>> true

Entanglement and Measurement

OQaml highlights the fact that a measurement is a gate operation, despite it being non-unitary. In general a measurement is connected to the (partial) collapse of a wave-function. It will factorize the wave-function with respect to the measured qubits. To understand what is happening let us construct a circuit to create the state

$$ |\Psi\rangle \sim (|00\rangle + |11\rangle)(|0\rangle + |1\rangle) $$

We can do this with a simple circuit

let tqvm = Q.apply (Q.CIRCUIT [Q.H 2; Q.CNOT (0,1); Q.H 0]) (Q.init_qvm 3);;

Measuring Qubit 2 with the MEASURE gate

let cqvm = Q.apply (Q.MEASURE 2) tqvm;;

>>> {Q.num_qubits = 3;
wf =
               C0
R0        (0, 0i)
R1 (0.707107, 0i)
R2        (0, 0i)
R3        (0, 0i)
R4        (0, 0i)
R5        (0, 0i)
R6        (0, 0i)
R7 (0.707107, 0i);

reg = [|0; 0; 1|]}

then collapses the state to either

$$ |\Phi_0\rangle \sim (|00\rangle + |11\rangle)|0\rangle $$

or

$$ |\Phi_1\rangle \sim (|00\rangle + |11\rangle)|1\rangle $$

leaving an entangled bell pair behind. On the other hand measuring Qubit 0 destroys the entanglement and results in

$$ |\tilde\Phi_0\rangle \sim |00\rangle (|0\rangle + |1\rangle) $$

or

$$ |\tilde\Phi_1\rangle \sim |11\rangle (|0\rangle + |1\rangle) $$

Note that the measurement returns a valid QVM, with the classic register being filled with the corresponding measurement outcome. The validity is guaranteed by the type definition of `MEASURE` and the type signature of the `apply` function. Both of the measurement tests described above can be confirmed using the `measure_all` functionality that let's you sample from a prepared QVM state: 
Q.measure_all cvqm 10;;

>>> [[0; 0; 1]; [1; 1; 1]; [1; 1; 1]; [1; 1; 1]; [0; 0; 1];
     [1; 1; 1]; [1; 1; 1]; [1; 1; 1]; [0; 0; 1]; [0; 0; 1]]

In this example we prepared the QVM in the cqvm state above and asked the system to sample 10 individual bitstrings from cqvm. This enables you to now do statistical analysis on these strings and infer information about the quantum system; one notable fact is that the first two bits are always identical hinting at the presence of entanglement.