Introduction into Quantum Computations Alexei Ashikhmin Bell Labs

Transcription

1 Introduction into Quantum Computations Alexei Ashikhmin Bell Labs Workshop on Quantum Computing and its Application March 16, 2017 Qubits Unitary transformations Quantum Circuits Quantum Measurements Quantum Fourier Transform Phase Estimation Order Finding Fast Factoring Quantum Error Correction

2 Dirac vs. Linear Algebra Notation is the inner product If then

3 Dirac vs. Linear Algebra Notation (cntd) Vectors Hence can be written in the form is an orthonormal basis of We can also say and write is an orthonormal basis of Typically we assume

4 Qubits Laser beams Electron can be in two states: Ground (G) 0> and Excited 1> Laser 1: electron moves from G to E state Laser 2: electron moves from E to G state Laser 3: electron moves to a superposition of states, e.g. 30% G and 70 % E Postulate 1: Pure state of a qubit is In our example

5 n Qubits qubits Postulate 1 The state (pure) of qubits is a vector hence, manipulating by qubits, we effectively manipulate by complex coefficients As a result we obtain a significant (sometimes exponential) speed up

6 Unitary Evolution Postulate 2 The time evolution of a closed quantum system is described by the Schrodinger equation is the system Hamiltonian, The solution of this equation is is unitary operator

7 Unitary Evolution Postulate 2 The time evolution of a closed quantum system is described by a unitary transformation Apply a unitary rotation state new state

8 Quantum Circuits Quant Not Gate LA notation:

9 Quantum Circuits Controlled Not Gate (Quant XOR Gate) two qubits in the joint state: the same qubits (particles) but in a new state: This circuit computes the Boolean function for ALL inputs simultaneously!

10 Quantum Circuits Classical AND and NOT gates form a universal set, i.e., they allow one to implement any Boolean function Hadamard, Phase, CNOT, and gates form a universal set, i.e., they allow one to approximate any unitary with arbitrary precision Can we approximate any given unitary using a circuit of size polynomial in? NO!

11 von Neumann Measurement and orthogonal subspaces; they span is the orthogonal projection on is the orthogonal projection on Postulate 3 is projected on with probability is projected on with probability We know to which subspace was projected quant. output classical output shows to which subspace or the state was projected

12 Quantum Fourier Transform Discrete Fourier Transform (DFT) of size N is defined as Quantum DFT is defined by QDFT or QDFT

13 Let and Quantum Fourier Transform After some computations one gets QDFT The final state is a tensor product of individual states of n qubits. Typically this means that it is not difficult to construct a quantum circuit for it.

14 Example. QDFT Circuit for Quantum Fourier Transform

15 The complexity of QDFT is Quantum Fourier Transform The complexity of Classical DFT is The complexity of Classical DFT that finds (with possible error) the largest coefficient of DFT is Can we use QDFT to get coefficients in NO!

16 Phase Estimation Let be a unitary operator and its eigenvector. So We would like to find the phase Phase Estimation has multiple applications

17 Phase Estimation Inverse QDFT

18 Order Finding x and N are coprime. We need to find the smallest r s Example No classical algorithms with complexity Quantum approach. Take s.t. For, we have

19 is s.t. Order Finding For, we have Theorem is unitary with eigenvalues and eigenvectors: and bits

20 We use this Order Finding in the phase estimation circuit with input Inverse QDFT At the output we get for random

21 Fast Factoring Finding the Greatest Common Devisor (gcd) of integers z and N is easy. Complexity Algorithm 1. Take random 2. Find its order, i.e., the smallest s.t. 3. If a. is even b. then find Theorem Either is a nontrivial factor of N or (and)

22 Quantum Errors Quantum computer is unavoidably vulnerable to errors Any quantum system is not completely isolated from the environment Uncertainty principle we can not simultaneously reduce: laser intensity and phase fluctuations magnetic and electric fields fluctuations momentum and position of an ion The probability of spontaneous emission is always greater than 0 Leakage error electron moves to a third level of energy

23 Depolarizing Channel (Standard Error Model) Depolarizing Channel means the absence of error are the flip, phase, and flip-phase errors respectively This is an analog of the classical quaternary symmetric channel

24 No-Cloning Theorem Perhaps the simplest classical error correcting code is repetition code. Encoding: , So if say 2 bits are flipped (01001) we still can say it was 0 Can we use the same idea for quantum error protection? No. We have a qubit in unknown state We bake our own qubit in any desirable state, say The joint state of is Theorem (No-Cloning) There is no unitary transform s.t.

25 Quantum Codes 1 2 k k+1 n unitary rotation 1 2 n information qubits in state the joint state: redundant qubits quantum codeword in the state, is a linear subspace of, is the code rate is an [[n,k]] quantum code

26 Classical Linear Codes Def. Binary linear [n,k] code is a k-dimensional subspace of (all summations and multiplications by modulo 2) Example. [5,2] code with code vectors: is its generator, k x n, matrix, its rows are basis vectors is its parity check, (n-k) x n, matrix The minimum distance of this code is d=5 We always have, where is an inner product

27 Quantum Stabilizer Codes is symplectic inner product: is a linear [2n, n-k] code, with and, is the dual code with and. If then is self-orthogonal (with respect to symplectic product) A self-orthogonal defines a stabilizer [[n,k]] quant. code

28 My Work on Quantum Codes Bounds Bounds on the tradeoffs between the code rate and error correction capabilities Bounds on the probability of undetected error Bounds on the probability of error in quantum Hybrid ARQ Bounds on Entanglement Assisted quantum stabilizer codes Constructions General construction of nonbinary quantum stabilizer codes BCH type quantum stabilizer codes Asymptotically good quantum codes with small construction complexity Quantum Codes Robust to Decoding Errors (DS codes) Bounds, Constructions, Performance of Radom DS codes

29 Bounds on the Minimum Distance of Quantum Codes New Upper bounds Knill, Laflamme s Singleton bound Existence bound

30 Robust Quantum Syndrome Measurement Msrmnt of g Msrmnt of g 2 3 Msrmnt of g 3 4 Msrmnt of g 4 5 Msrmnt of g 1 12 Msrmnt of g 4 5 s (1) s (1) 2 s (2) 1 s (1) s (1) s (3) 4 5 qubits in the state, is a code vector of [[5,1]] code

31 Robust Quantum Syndrome Measurement Msrmnt of g 1 2 Msrmnt of g 2 3 Msrmnt of g 3 4 Msrmnt of g 4 5 Msrmnt of f 1 12 Msrmnt of f 8 5 Decoder of classical [12,4] code 5 qubits in the state, is a code vector of [[5,1]] code

32 Thank You