Quantum Computing Michael Larson

The Quantum Computer Quantum computers, like all computers, are machines that perform calculations upon data. Quantum computers use the principles of superposition and entanglement from the field of quantum mechanics to aid in the performance of these calculations

Superpositioning Superpositioning is a fundamental principle of quantum mechanics Combinations of valid quantum states are themselves valid quantum states An unobserved particle that is capable of being measured in one of two states exists as a superposition of those states

Measurement and Decoherence Superpositioning only occurs while the system is not being measured When a system is measured, it’s superposition collapses to a well defined state. A superposition can be described as the sum of the probabilities that a measurement will find a system in a certain state. Accidental “measurement” of the quantum state leads to unintended collapse of the superposition.

Quantum Entanglement Quantum entanglement occurs when a group of quantum particles cannot be described independently. For quantum computers this means that modification of a quantum bit or register will project changes onto their entangled bits or registers For example, assume “The sum of these two bits is always 1” is a known fact. 1010

The Quantum Bit In a regular computer a bit exists in one of two states, either 0 or 1. In a quantum computer a quantum bit, or qubit exists as a superposition of these two states The creation and manipulation of these quantum bits is the objective of quantum computing

Registers of Qubits A register containing n qubits can be expressed mathematically as a superposition of 2 n quantum states. These states can be described with a 2 n dimensional vector with complex coefficients representing the probability of the represented state. The sum of squares of the magnitude of these coefficients must be equal to 1. Superpositioning and subsequent collapse upon measurement means that quantum computing is nondeterministic. Many samples should be taken to build a probability distribution when calculating an answer

Example 3-Qubit Register StateCoefficientProbability 000a|a| 2 001b|b| 2 010c|c| 2 011d|d| 2 100e|e| 2 101f|f| 2 110g|g| 2 111h|h| 2 Sum of Probabilities |a| 2 + |b| 2 + |c| 2 + |d| 2 + |e| 2 + |f| 2 + |g| 2 + |h| 2 = 1

Shor’s Algorithm The objective of Shor’s algorithm is find a factor of a large number N. Repeatedly finding factors and dividing will result in finding all factors of N. The proposed quantum algorithm works much faster than the best classical algorithm for integer factorization Difficulty and time complexity of large integer factorization is the basis of many security schemes.

Classical View of Shor’s Algorithm 1. Pick a random number a<N and see if it shares a greatest common denominator with N. 2. If the GCD is greater than 1, that GCD is a factor and we are done 3. Use the quantum portion of Shor’s algorithm to find the period of f(x) = a x mod N 4. After finding the smallest period r such that f(x)= f(x+r), this period can be used to generate a r/2 +1 and a r/2 -1, both of which are likely to have a greatest common denominator with N. 5. If not go back to step 1. Step 3 is the quantum portion of this algorithm

Shor’s Period Finding Subroutine Classically we need to calculate a x mod N for every x<N, then analyze the pattern produced to find the period r. For very large N, this very computationally and memory intensive from a classical standpoint. Writing the algorithm as a quantum algorithm that takes advantage of the principles of entanglement and superposition would allow for massive reduction in data and computational cost

Shor’s Period Finding Subroutine(cont) With sufficient qubits, we can initialize a register as a superposition of every 0<=x <= N. The calculation a x mod N can be performed using quantum gates, projecting this distribution onto the second register. Due to effects of constructive and destructive interference in quantum algorithms, certain values of x that seperated by distance r tend to interfere constructively, while others tend to interfere destructively. This leads to one state in the 2nd register having a much higher probability of being measured than any other.

Shor’s Period Finding Subroutine(cont) Due to the principles of quantum entanglement, each time a result is measured in the second register, the first register collapses to a state congruent with that state. If a x mod N is measured to be 1, the probability will be evenly distributed in the first register among x’s such that a x mod N = 1. Given enough sampling, a set of states separated by r would be most commonly measured in register 1 after running the algorithm and collapsing register 2

Shor’s Period Finding Algorithm Example xaxax a x mod N = m x (cont)a x (cont)a x mod N =m (cont)

Shor’s Period Finding Algorithm Example Suppose that measuring the second register most commonly collapses to m = 8. Entanglement with the x register then collapses the register like so (x states not shown have 0 probability) xaxax a x mod N = m

Results Repeated experiments and measuring of the X register will lead to a distribution of elements that are multiples of 4 apart from each other Thus r = 4 N = 15, a = 2 a r/2 + 1 = 5, a r/2 - 1 = 3, both of which are supposed share a greatest common denominator with N=15. In fact 3 and 5 are verifiably the only 2 factors of 15.

Relevance to Parallelism Shor’s algorithm shows that with enough quantum bits and the right algorithm a quantum computer can run massively parallel code by superpositioning various possible initial states and collapsing them with measurement. Every quantum bit effectively doubles parallelism of quantum code because it doubles the number of superpositioned states.

Questions?* *answered only to the best of my understanding.