Fault-tolerance Quantum computation and Quantum Error Correction

Editor's Notes

• #4: A quantum computer is a machine that relies on quantum phenomena like superposition, quantum uncertainty, and entanglement. SUPERPOSITION: there is an equal probability that something is either in one state (1) or another (0). Thus, something is in both states, or between both states at the same time until observedQuantum Uncertainty: states that the position and velocity of a particle are unknown until observed. Quantum Nonlocality—Entanglement: When two particles share the same quantum state they are entangled. This means that two or more particles will share the same properties: for example, their spins are related. Even when removed from each other, these particles will continue to share the same properties. We can say: Information is stored in a physical medium, and manipulated by physical processes. Therefore, the laws of physics dictate the capabilities of any information processing device. As we know, classical physics is incomplete to explain some physical events especially on the atomic and subatomic level and has been replaced by a more powerful framework: quantum mechanics. Consequently, a computer that operates on quantum states can perform tasks that are beyond the capability of any conceivable classical computer.
• #5: Google announced it has a quantum computer that is 100 million times faster than any classical computer in its lab. For example, a quantum computer with 30 qubits equals the processing power of conventional computer that running at 10 teraflop (trillions of floating-point operations per second). Not only is this expected to make quantum computers faster and more efficient than even the most powerful current supercomputer, it’s thought they could have potential uses and solve problems that we can’t even comprehend yet because of quantum phenomena like quantum superposition and entanglement. Quantum computer are good at solving certain classes of problems like factoring large primes, random walks, unordered data retrieval, etc. Not good at things like: exact answers, logic, pattern recognition, unconstrained control problems.
• #7: In 2016, IBM was the first company to put a quantum computer on the cloud. The company has since built up an active community of more than 260,000 registered users, who run more than one billion every day on real hardware and simulators. In 2017, IBM was the first company to offer universal quantum computing systems via the IBM Q Network. Intel designed and fabricated the new 49-qubit superconducting quantum chip in 2018. Founded in 1999, D-Wave claims to be the first company to sell a commercial quantum computer, in 2011, and the first to give developers real-time cloud access to quantum processors with Leap, its quantum cloud service. D-Wave's approach to quantum computing, known as quantum annealing, is best suited to optimization tasks. In nature, physical systems tend to evolve toward their lowest energy state: objects slide down hills, hot things cool down, and so on. This behavior also applies to quantum systems. To imagine this, think of a traveler looking for the best solution by finding the lowest valley in the energy landscape that represents the problem.  Google built a quantum computer with 53 qubits.
• #8: This picture presents IBM’s roadmap to advance quantum computers from today’s noisy, small-scale devices to larger, more advance quantum systems of the future. Currently the company's researchers are developing a suite of scalable, increasingly larger and better processors, with a 1,000-plus qubit device - called IBM Quantum Condor - targeted for the end of 2023.
• #9: In quantum computing the information is encoded in Quantum bits (or qubits), which are somewhat analogous to the bit in classical computation. This information is described by a state in a 2-level quantum mechanical system which is formally equivalent to a two-dimentional vector space over the complex numbers. The two basis states are conventionally written as ∣0⟩ and ∣1⟩ (pronounced: 'ket 0' and 'ket 1'), this notation is called Dirac notation. A qubit can be in ∣0⟩, ∣1⟩ or (unlike a classical bit) in a linear combination of both states. The name of this phenomenon is superposition. Superposition: there is an equal probability that something is either in one state (1) or another (0). Thus, something is in both states, or between both states at the same time until observed. Where α and β are complex numbers and are called quantum amplitudes. A qubit in superposition is in both of the states at the same time with probabilities. In other words, 𝛼 2 : tells us the probability of finding ∣ψ⟩ in state ∣0⟩ and 𝛽 2 : tells us the probability of finding ∣ψ⟩ in state ∣1⟩.
• #12: It turns out that we can describe the interaction of a single qubit with the environment and hence single-qubit errors with the familiar operators I,X,Y,Z. The identity operator, of course, represents no error at all. In the case of a classical bit, which can be 0 or 1, engineers are concerned with bit flip errors. The physical details aren’t important for us, but we can imagine some stray electromagnetic field causing a bit to change from 1 to 0, for example. A similar type of error can affect qubits, where we have |0>→|1> and |1>→|0>. This type of error is described by the X operator, In quantum systems, bit flip errors are not the only problem that we can encounter. We can also have phase flip errors. we readily see that a phase flip error is described by the Z Pauli operator. We recall that Z acts on a qubit in the following way: The Y operator is related to a phase flip followed by a bit flip. Later we will see how quantum error-correcting codes work on bit flip and phase flip errors
• #19: The repetition code works in a classical channel, because classical bits are easy to measure and to repeat. Repetition code in quantum is not possible because of no-cloning theorem. A different method, such as the so-called three-qubit bit flip code, has to be used. This technique uses entanglement and syndrome measurements and is comparable in performance with the repetition code. It is important to note that the state |ψ> is not triplicated but only the basis vectors are triplicated. This redundancy makes it possible to detect errors in |ψL> and correct them as we see in the next figure.
• #20: The gate NX stands for the bit-flip noise Now the state |ψL> is sent through a quantum channel which introduces bitflip error with a rate p for each qubit independently. We assume p is sufficiently small so that not many errors occur during the qubit transmission. The received state depends on in which physical qubit(s) the bit-flip occurred. Two ancillary qubits are perpared in the state |00> as depicted in Fig and applies four CNOT operations whose control bits are the encoded qubits while the target qubits are two ancillary qubits. Let |x1x2x3 > be a basis vector has received and let A (B) be the output state of the first (second) ancilla qubit. (A = x1 ⊕ x2 and B = x1 ⊕ x3)
• #21: The gate NX stands for the bit-flip noise Now the state |ψL> is sent through a quantum channel which introduces bitflip error with a rate p for each qubit independently. We assume p is sufficiently small so that not many errors occur during the qubit transmission. The received state depends on in which physical qubit(s) the bit-flip occurred. Two ancillary qubits are perpared in the state |00> as depicted in Fig and applies four CNOT operations whose control bits are the encoded qubits while the target qubits are two ancillary qubits. Let |x1x2x3 > be a basis vector has received and let A (B) be the output state of the first (second) ancilla qubit. (A = x1 ⊕ x2 and B = x1 ⊕ x3)
• #23: The gate NX stands for the bit-flip noise Now the state |ψL> is sent through a quantum channel which introduces bitflip error with a rate p for each qubit independently. We assume p is sufficiently small so that not many errors occur during the qubit transmission. The received state depends on in which physical qubit(s) the bit-flip occurred. Two ancillary qubits are perpared in the state |00> as depicted in Fig and applies four CNOT operations whose control bits are the encoded qubits while the target qubits are two ancillary qubits. Let |x1x2x3 > be a basis vector has received and let A (B) be the output state of the first (second) ancilla qubit. (A = x1 ⊕ x2 and B = x1 ⊕ x3)
• #24: The gate NX stands for the bit-flip noise Now the state |ψL> is sent through a quantum channel which introduces bitflip error with a rate p for each qubit independently. We assume p is sufficiently small so that not many errors occur during the qubit transmission. The received state depends on in which physical qubit(s) the bit-flip occurred. Two ancillary qubits are perpared in the state |00> as depicted in Fig and applies four CNOT operations whose control bits are the encoded qubits while the target qubits are two ancillary qubits. Let |x1x2x3 > be a basis vector has received and let A (B) be the output state of the first (second) ancilla qubit. (A = x1 ⊕ x2 and B = x1 ⊕ x3)
• #26: We consider how to correct phase-flip error mimicking the prescription given for a bit-flip error. We generate a logical qubit that takes α|000>+β|111>, and then apply Hadamard gates to each qubit.
• #30: An n–qubit quantum state is called a stabilizer state if it is stabilized by nontrivial subgroup of Pauli group. An important property of the Pauli group is that any two Pauli operators either commute or anticommute Based on this property we can see how a stabilizer code detects and corrects errors. A valid code word will be a +1 eigenvalue of all the stabilizer generators. Suppose that an error operator E will anticommute with some of the stabilizer generators, and commute with others. The error will be detected that it anticommute with some of the stabilizer generators.
• #37: As errors occur from the random and unpredictable appearance of X and Z operations, they must be detected by repeatedly measuring each qubit, which can be done with combined X and Z measurements.
• #46: A error gate, is called depolarizing noise, is an imperfection in any operation we perform. The effect of this will be, with probability Pgate ,to replace the state of any qubit with a completely random state. The other form of noise is that for measurement. This simply flips between a 0 to a 1 and vice-versa immediately before measurement with probability Pmeas. def get_noise(p_meas,p_gate):
• #47: A error gate, is called depolarizing noise, is an imperfection in any operation we perform. The effect of this will be, with probability Pgate ,to replace the state of any qubit with a completely random state. The other form of noise is that for measurement. This simply flips between a 0 to a 1 and vice-versa immediately before measurement with probability Pmeas. def get_noise(p_meas,p_gate):
• #48: A error gate, is called depolarizing noise, is an imperfection in any operation we perform. The effect of this will be, with probability Pgate ,to replace the state of any qubit with a completely random state. The other form of noise is that for measurement. This simply flips between a 0 to a 1 and vice-versa immediately before measurement with probability Pmeas. def get_noise(p_meas,p_gate):
• #49: A error gate, is called depolarizing noise, is an imperfection in any operation we perform. The effect of this will be, with probability Pgate ,to replace the state of any qubit with a completely random state. The other form of noise is that for measurement. This simply flips between a 0 to a 1 and vice-versa immediately before measurement with probability Pmeas. def get_noise(p_meas,p_gate):
• #50: With this noise, all outcomes occur with equal probability, with differences in results being due only to statistical noise. No trace of the encoded state remains. This is an important point to consider for error correction: sometimes the noise is too strong to be corrected. The optimal approach is to combine a good way of encoding the information you require, with hardware whose noise is not too strong.
• #53: In these circuits, we have two types of physical qubits. There are the 'code qubits', which are the three physical qubits across which the logical state is encoded. There are also the 'link qubits', which serve as the auxiliary qubits for the syndrome measurements. Our single round of syndrome measurements in these circuits consist of just two syndrome measurements. One compares code qubits 0 and 1, and the other compares code qubits 1 and 2. One might expect that a further measurement, comparing code qubits 0 and 2, should be required to create a full set. However, these two are sufficient. This is because of the information on whether 0 and 2 have the same z basis state can be inferred from the same information about 0 and 1 with that for 1 and 2. Indeed, for n qubits, we can get the required information from just n−1 syndrome measurements of neighbouring pairs of qubits.