computing environment without any noise were to be obtained, there is still the natural property of qubits to decay that must be addressed. The excited state of qubits ultimately decays to the ground state due to being coupled to the vacuum fluctuation (the vacuum fluctuations of electromagnetic fields are unavoidable). Therefore, it is necessary to encode the quantum states to be protected in such a way that they can be error-corrected to remain robust. These requirements introduce the notions of qubit lifecycle and qubit management in quantum information systems.
4.4.3 Entanglement property of qubits
Unlike classical information, which can be examined arbitrarily many times to determine if it has changed, quantum information cannot be measured directly because it will change or destroy the information. However, quantum information has the interesting property of entanglement. The entanglement property refers to quantum particles being entangled with one another. Quantum particles are not isolated and discrete, but rather correlated, both with each other, and the history of the system. Particles and previous states are correlated as part of the fuller information landscape of the quantum system. Hence, quantum error correction is performed by taking advantage of the entanglement property of qubits. The insight is that due to entanglement, it is possible to measure relationships between qubits without measuring the values stored by the qubits themselves.
4.4.3.1Error-correction codes
Quantum error correction uses the entanglement property to smear out the information of 1 qubit onto 9 entangled qubits (in the basic case). The idea is that the information of 1 qubit can be stored on a highly entangled state of several qubits. A local point, 1 qubit, can be smeared out, and represented with a larger number of qubits. The auxiliary qubits are called an ancilla (ancillary qubits). Quantum error correction is typically performed by introducing an ancilla in the form of additional qubits with which the qubit of interest is entangled. Entangling the qubit of interest with ancillary qubits allows the qubit to be protected by smearing out its information over a larger system. The qubit of interest can be error-checked since its information can be examined indirectly through the entangled qubits (through parity measurements). In this way, quantum error correction is used to test the integrity of quantum information without damaging it. Quantum error correction makes quantum computing feasible, in that a quantum computer can tolerate some degree of noise in the environment by correcting errors. Many different kinds of quantum error-correction codes (encoding schemes) have been proposed. Shor’s code uses a 9-qubit smear, and others require fewer qubits (a 7-qubit code and a 5-qubit code, for example) (Shor, 1995).
4.4.3.2Classical error correction
In general, all forms of computing systems use error correction to check for data integrity. An error-correcting process seeks to determine whether information has been damaged or destroyed and restores its initial state. Error correction is well-understood in classical logic. For example, there could be a memory chip storing bits that is hit by a cosmic ray. If one bit in a 32-bit word is flipped, there are many known ways of recovery. One frequently used method is having many copies of the data. With redundancy, having several copies of the information means that a mechanistic majority-voting mechanism can be used to confirm the intact version of the data. With quantum logic, however, error correction based on making redundant copies of the same information cannot work. It is not possible to copy quantum information due to the no-cloning theorem, which states that it is impossible to create an identical copy of an arbitrary unknown quantum state. Therefore, quantum error-correction methods such as those based on entanglement are needed.
4.4.3.3Shor’s code
It is not by accident that Shor’s code, the first quantum error-correction code discovered, is 9 qubits. Nine qubits is the smallest ancilla that can be used to confirm that the original qubit was not flipped (changed or damaged), by checking various pairwise sequences of possible flipping along the X, Y, and Z axes of the qubit. A simple error-correcting code could instantiate a single logical qubit of data as three physical qubits for each scenario of the three axes. With pair-wise evaluation, it is possible to determine whether the first and second qubits have the same value, and whether the second and third qubits have the same value, without determining what that value is. If one of the qubits turns out to disagree with the other two, it can be reset to their value. Further, the pair-wise evaluations might be performed in both time and space suggesting quantum information processing architectures with time speed-ups.
Shor’s code is a sequential method using single Pauli operators (3D operators with X, Y, and Z values) to act on the system according to the different possible error permutations that could have occurred. Since the ancillary qubits and the original qubit are entangled, any error will have a recognizable signature and can be corrected (by repairing it into the initial phase or into an irrelevant phase that does not impact the original qubit’s information). Since the states of the system are eigenstates of eigenvalue one for all of the operators, the measurement does nothing to the overall state. The Shor code is redundant, in that the number of bits of information it protects is significantly fewer than the number of physical bits that are present. Noise can come in from the environment without disrupting the message. Also, the Shor code is nonlocal in the sense that the qubit information is carried in the entanglement between the multiple qubits that protect it against local decoherence and depolarization.
The Pauli operator that uses X–Y–Z quantum spin representations is one proposed method. Kraus operators, which are operator-sum representations, are another (Verlinde & Verlinde, 2013). However, Kraus operators can be difficult to engage because they require details of the interaction with the environment, which may be unknown. Single Pauli operators suggest a more straightforward implementation model.
4.4.4 Quantum information processors
The main concept of error correction is that to protect qubits from environmental noise and to mitigate against state decay, the qubit of interest can be encoded in a larger number of ancillary qubits through entanglement. The entangled qubits are combined into a bigger overall fabric of qubits that constitutes the quantum information processor. For example, a quantum information processor might have 50 qubits, and an error-correction requirement of 9 qubits for each qubit. This would only leave 5 qubits available for information processing. The scaling challenge is clear, in that with current error-correction methods, most of the processing capacity must be devoted to error correction. The implied scaling rule is 10, in the sense that any size quantum information processor only has available one-tenth of the total qubits for the actual information processing (each 1 qubit requires 9 qubits of error correction). More efficient error-correcting codes have been proposed, but the scaling rule of 10 could be a general heuristic, at least initially.
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