Deutsch–Jozsa algorithm

Classical solution

For a conventional deterministic algorithm where ${\displaystyle n}$ is the number of bits, ${\displaystyle 2^{n-1}+1}$ evaluations of ${\displaystyle f}$ will be required in the worst case. To prove that ${\displaystyle f}$ is constant, just over half the set of inputs must be evaluated and their outputs found to be identical (because the function is guaranteed to be either balanced or constant, not somewhere in between). The best case occurs where the function is balanced and the first two output values are different. For a conventional randomized algorithm, a constant ${\displaystyle k}$ evaluations of the function suffices to produce the correct answer with a high probability (failing with probability ${\displaystyle ϵ\le 1/2^k}$ with ${\displaystyle k\ge 1}$). However, ${\displaystyle k=2^{n-1}+1}$ evaluations are still required if we want an answer that has no possibility of error. The Deutsch-Jozsa quantum algorithm produces an answer that is always correct with a single evaluation of ${\displaystyle f}$.

History

The Deutsch–Jozsa algorithm generalizes earlier (1985) work by David Deutsch, which provided a solution for the simple case where ${\displaystyle n=1}$.
Specifically, finding out if a given Boolean function whose input is one bit, ${\displaystyle f:\{0,1\}\to \{0,1\}}$, is constant.

The algorithm, as Deutsch had originally proposed it, was not deterministic. The algorithm was successful with a probability of one half. In 1992, Deutsch and Jozsa produced a deterministic algorithm which was generalized to a function which takes ${\displaystyle n}$ bits for its input. Unlike Deutsch's algorithm, this algorithm required two function evaluations instead of only one.

Further improvements to the Deutsch–Jozsa algorithm were made by Cleve et al., resulting in an algorithm that is both deterministic and requires only a single query of ${\displaystyle f}$. This algorithm is still referred to as Deutsch–Jozsa algorithm in honour of the groundbreaking techniques they employed.

Algorithm

For the Deutsch–Jozsa algorithm to work, the oracle computing ${\displaystyle f(x)}$ from ${\displaystyle x}$ must be a quantum oracle which does not decohere ${\displaystyle x}$. In its computation, it cannot make a copy of ${\displaystyle x}$, because that would violate the no cloning theorem. The point of view of the Deutsch-Jozsa algorithm of ${\displaystyle f}$ as an oracle means that it does not matter what the oracle does, since it just has to perform its promised transformation.

The algorithm begins with the ${\displaystyle n+1}$ bit state ${\displaystyle |0{⟩}^{\otimes n}|1⟩}$. That is, the first n bits are each in the state ${\displaystyle |0⟩}$ and the final bit is ${\displaystyle |1⟩}$. A Hadamard gate is applied to each bit to obtain the state

$${\displaystyle \frac{1}{\sqrt{2^{n+1}}}\sum _{x=0}^{2^n-1}|x⟩(|0⟩-|1⟩),}$$

where ${\displaystyle x}$ runs over all ${\displaystyle n}$-bit strings, which each may be represented by a number from ${\displaystyle 0}$ to ${\displaystyle 2^n-1}$. We have the function ${\displaystyle f}$ implemented as a quantum oracle. The oracle maps its input state ${\displaystyle |x⟩|y⟩}$ to ${\displaystyle |x⟩|y\oplus f(x)⟩}$, where ${\displaystyle \oplus }$ denotes addition modulo 2. Applying the quantum oracle gives:

$${\displaystyle \frac{1}{\sqrt{2^{n+1}}}\sum _{x=0}^{2^n-1}|x⟩(|0\oplus f(x)⟩-|1\oplus f(x)⟩).}$$

For each ${\displaystyle x,f(x)}$ is either 0 or 1. Testing these two possibilities, we see the above state is equal to

$${\displaystyle \frac{1}{\sqrt{2^{n+1}}}\sum _{x=0}^{2^n-1}(-1{)}^{f(x)}|x⟩(|0⟩-|1⟩).}$$

At this point the last qubit ${\displaystyle \frac{|0⟩-|1⟩}{\sqrt{2}}}$ may be ignored and the following remains:

$${\displaystyle \frac{1}{\sqrt{2^n}}\sum _{x=0}^{2^n-1}(-1{)}^{f(x)}|x⟩.}$$

Next, we will have each qubit go through a Hadamard gate. The total transformation over all ${\displaystyle n}$ qubits can be expressed with the following identity:

$${\displaystyle H^{\otimes n}|k⟩=\frac{1}{\sqrt{2^n}}\sum _{j=0}^{2^n-1}(-1{)}^{k\cdot j}|j⟩}$$

(${\displaystyle j\cdot k=j_0{k}_0\oplus j_1{k}_1\oplus \cdots \oplus j_{n-1}{k}_{n-1}}$ is the sum of the bitwise product). This results in

$${\displaystyle \frac{1}{\sqrt{2^n}}\sum _{x=0}^{2^n-1}(-1{)}^{f(x)}\left[\frac{1}{\sqrt{2^n}}\sum _{y=0}^{2^n-1}{\left(-1\right)}^{x\cdot y}|y⟩\right]=\sum _{y=0}^{2^n-1}\left[\frac{1}{2^n}\sum _{x=0}^{2^n-1}(-1{)}^{f(x)}(-1{)}^{x\cdot y}\right]|y⟩.}$$

From this, we can see that the probability for a state ${\displaystyle k}$ to be measured is

$${\displaystyle {\left|\frac{1}{2^n}\sum _{x=0}^{2^n-1}{\left(-1\right)}^{f(x)}{\left(-1\right)}^{x\cdot k}\right|}^2}$$

The probability of measuring ${\displaystyle k=0}$, corresponding to ${\displaystyle |0{⟩}^{\otimes n}}$, is

$${\displaystyle |\frac{1}{2^n}\sum _{x=0}^{2^n-1}(-1{)}^{f(x)}{|}^2}$$

which evaluates to 1 if ${\displaystyle f(x)}$ is constant (constructive interference) and 0 if ${\displaystyle f(x)}$ is balanced (destructive interference). In other words, the final measurement will be ${\displaystyle |0{⟩}^{\otimes n}}$ (all zeros) if and only if ${\displaystyle f(x)}$ is constant and will yield some other state if ${\displaystyle f(x)}$ is balanced.

Deutsch's algorithm

Deutsch's algorithm is a special case of the general Deutsch–Jozsa algorithm where n = 1, so that ${\displaystyle f:\{0,1\}\to \{0,1\}}$. We need to check the condition ${\displaystyle f(0)=f(1)}$. It is equivalent to check ${\displaystyle f(0)\oplus f(1)}$ (where ${\displaystyle \oplus }$ is addition modulo 2, which can also be viewed as a quantum XOR gate implemented as a Controlled NOT gate), if zero, then ${\displaystyle f}$ is constant, otherwise ${\displaystyle f}$ is not constant.

We begin with the two-qubit state ${\displaystyle |0⟩|1⟩}$ and apply a Hadamard gate to each qubit. This yields $${\displaystyle \frac{1}{2}(|0⟩+|1⟩)(|0⟩-|1⟩).}$$

We are given a quantum implementation of the function ${\displaystyle f}$ that maps ${\displaystyle |x⟩|y⟩}$ to ${\displaystyle |x⟩|f(x)\oplus y⟩}$. Applying this function to our current state we obtain $${\displaystyle \begin{array}{cc}& \frac{1}{2}(|0⟩(|f(0)\oplus 0⟩-|f(0)\oplus 1⟩)+|1⟩(|f(1)\oplus 0⟩-|f(1)\oplus 1⟩))\\ & =\frac{1}{2}((-1{)}^{f(0)}|0⟩(|0⟩-|1⟩)+(-1{)}^{f(1)}|1⟩(|0⟩-|1⟩))\\ & =(-1{)}^{f(0)}\frac{1}{2}\left(|0⟩+(-1{)}^{f(0)\oplus f(1)}|1⟩\right)(|0⟩-|1⟩).\end{array}}$$

We ignore the second qubit and the global phase and therefore have the state $${\displaystyle \frac{1}{\sqrt{2}}(|0⟩+(-1{)}^{f(0)\oplus f(1)}|1⟩).}$$

Applying a Hadamard gate to this state we have $${\displaystyle \begin{array}{cc}& \frac{1}{2}(|0⟩+|1⟩+(-1{)}^{f(0)\oplus f(1)}|0⟩-(-1{)}^{f(0)\oplus f(1)}|1⟩)\\ & =\frac{1}{2}((1+(-1{)}^{f(0)\oplus f(1)})|0⟩+(1-(-1{)}^{f(0)\oplus f(1)})|1⟩).\end{array}}$$

${\displaystyle f(0)\oplus f(1)=0}$ if and only if we measure ${\displaystyle |0⟩}$ and ${\displaystyle f(0)\oplus f(1)=1}$ if and only if we measure ${\displaystyle |1⟩}$. So with certainty we know whether ${\displaystyle f(x)}$ is constant or balanced.

Deutsch–Jozsa algorithm Qiskit implementation

The quantum circuit shown here is from a simple example of how the Deutsch–Jozsa algorithm can be implemented in Python using Qiskit, an open-source quantum computing software development framework by IBM.