Instruct: Given a question, retrieve passages that can help answer the question.
Query: What is a qubit?
In quantum computing, a qubit(/ˈkjuːbɪt/)or quantum bit(sometimes qbit) is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include: the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superpositionof both states/levels simultaneously, a property which is fundamental to quantum mechanics and quantum computing.
An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit quantum entanglement. Quantum entanglement is a nonlocal property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems.
, most technologies used to implement qubits face issues of stability, decoherence, fault tolerance and scalability. Because of this, many physical qubits are needed for the purposes of error-correction to produce an entity which behaves logically as a single qubit would in a quantum circuit or algorithm; this is the subject of quantum error correction. Thus, contemporary logical qubits typically consist of many physical qubits to provide stability, error-correction and fault tolerance needed to perform useful computations.
Quantum circuit for a single qubit quantum teleportation is demonstrated using Q-Kit, a quantum circuit simulator with graphical interface. Alice's state in qubit 2 is transferred to Bob's qubit 0 using a priorly entangled pair of qubits between Alice and Bob, qubits 1 and 0.
The CNOT gate operates on a quantum register consisting of 2 qubits. The CNOT gate flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is formula_1.
Coupling qubits is essential for implementing 2-qubit gates. Coupling two qubits may be achieved by connecting them to an intermediate electrical coupling circuit. The circuit might be a fixed element, such as a capacitor, or controllable, such as a DC-SQUID. In the first case, decoupling the qubits (during the time the gate is off) is achieved by tuning the qubits out of resonance one from another, i.e. making the energy gaps between their computational states different. This approach is inherently limited to allow nearest-neighbor coupling only, as a physical electrical circuit is to be lay out in between the connected qubits. Notably, D-Wave Systems' nearest-neighbor coupling achieves a highly connected unit cell of 8 qubits in the Chimera graph configuration. Generally, quantum algorithms require coupling between arbitrary qubits, therefore the connectivity limitation is likely to require multiple swap operations, limiting the length of the possible quantum computation before the processor decoherence.
One popular gating mechanism includes two qubits and a bus, all tuned to different energy level separations. Applying microwave excitation to the first qubit, with a frequency resonant with the second qubit, causes a formula_21 rotation of the second qubit. The rotation direction depends on the state of the first qubit, allowing a controlled phase gate construction.
More formally, following the notation of, the drive Hamiltonian describing the system excited through the first qubit driving line is
This shows that information has been encoded into a "subspace" of the two-qubit Hilbert space. Another way of encoding the same information is to encode "only" one of the qubits of the two qubits. Suppose the first qubit is encoded, then the state of the second qubit is completely arbitrary since:
A number of qubits taken together is a qubit register. Quantum computers perform calculations by manipulating qubits within a register. A qubyte (quantum byte) is a collection of eight qubits.[6]
In a quantum computer, the fundamental unit of information (called a quantum bit or qubit), is not binary but rather more quaternary in nature. This qubit property arises as a direct consequence of its adherence to the laws of quantum mechanics, which differ radically from the laws of classical physics. A qubit can exist not only in a state corresponding to the logical state 0 or 1 as in a classical bit, but also in states corresponding to a blend or quantum superposition of these classical states. In other words, a qubit can exist as a zero, a one, or simultaneously as both 0 and 1, with a numerical coefficient representing the probability for each state. A quantum computer manipulates qubits by executing a series of quantum gates, each a unitary transformation acting on a single qubit or pair of qubits. In applying these gates in succession, a quantum computer can perform a complicated unitary transformation to a set of qubits in some initial state.
The irreversible case is an extension of the reversible case: it uses the reversible algorithm as a subroutine. The irreversible algorithm contains another procedure called "Refresh" and extends the reversible one by using a heat bath. This allows for cooling certain qubits (called "reset qubits") without affecting the others, which results in an overall cooling of all the qubits as a system. The cooled reset qubits are used for cooling the rest (called "computational qubits") by applying a compression on them which is similar to the basic compression subroutine from the reversible case. The "insulation" of the computational qubits from the heat bath is a theoretical idealization that does not always hold when implementing the algorithm. However, with a proper choice of the physical implementation of each type of qubit, this assumption fairly holds.
Unlike quantum key distribution, quantum coin flipping is a protocol that is used between two participants who do not trust each other. The participants communicate via a quantum channel and exchange information through the transmission of qubits. For example, the sender, Alice, will determine a random basis and sequence of qubits and then transmit them to Bob. Bob then detects and records the qubits. Once Bob has recorded the qubits sent by Alice, he makes a guess to Alice on what basis she chose. Alice reports whether he won or lost to Bob and then sends Bob her entire original qubit sequence. Since the two parties do not trust each other, cheating is likely to occur at any step in the process.
The algorithm can be written using quantum operations on qubits, as opposed to the classical treatment. In particular, the C-NOT and C-SWAP steps can be replaced by a single unitary quantum operator that operates on the 3 qubits. Although this operation changes qubits formula_50 in a different manner than the two classical steps, it yields the same final bias for qubit formula_51. The operator formula_107 can be uniquely defined by its action on the computational basis of the Hilbert space of 3 qubits:
In matrix form, this operator is the identity matrix of size 8, except that the 4th and 5th rows are swapped. The result of this operation can be obtained by writing the product state of the 3 qubits, formula_116, and applying formula_107 on it. Afterwards, the bias of qubit formula_51 can be calculated by projecting its state on the state formula_1 (without projecting qubits formula_50) and taking the trace of the result (see density matrix measurement):
Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some specified operation. In particular, the controlled NOT gate (or CNOT or cX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is | 1 ⟩ {\displaystyle |1\rangle } , and otherwise leaves it unchanged. With respect to the unentangled product basis { | 00 ⟩ {\displaystyle \{|00\rangle } , | 01 ⟩ {\displaystyle |01\rangle } , | 10 ⟩ {\displaystyle |10\rangle } , | 11 ⟩ } {\displaystyle |11\rangle \}} , it maps the basis states as follows:
D-Wave Two boasts a CPU of about 512 qubits—an improvement over the D-Wave One series' CPUs of about 128 qubits The number of qubits can vary from chip to chip, due to variations in manufacturing. The increase in qubit count for the D-Wave Two was accomplished by tiling qubit pattern of the D-Wave One. This pattern, named "chimera" by D-Wave Systems, has a limited connectivity such that a given qubit can only interact with at most six other qubits. As with the D-Wave One, this restricted connectivity greatly limits the optimization problems that can be approached with the hardware.
The quantum circuits used for this algorithm are custom designed for each choice of formula_1 and each choice of the random formula_38 used in formula_57. Given formula_58, find formula_59 such that formula_60, which implies that formula_61. The input and output qubit registers need to hold superpositions of values from formula_62 to formula_63, and so have formula_48 qubits each. Using what might appear to be twice as many qubits as necessary guarantees that there are at least formula_1 different values of formula_66 that produce the same formula_67, even as the period formula_53 approaches formula_69.
The number of rounds is not bounded: since the biases of the reset qubits asymptotically reach the bias of the bath after each round, the bias of the target computational qubit asymptotically reaches its limit as the algorithm proceeds. The target qubit is the computational qubit that the algorithm aims to cool the most. The "cooling limit" (the maximum bias the target qubit can reach) depends on the bias of the bath and the number of qubits of each kind in the system. If the number of the computational qubits (excluding the target one) is formula_170 and the number of reset qubits is formula_171, then the cooling limit is formula_172. In the case where formula_173, the maximal polarization that can be obtained is proportional to formula_174. Otherwise, the maximal bias reaches arbitrarily close to formula_175. The number of rounds required in order to reach a certain bias depends on the desired bias, the bias of the bath and the number of qubits, and moreover varies between different versions of the algorithm.
Logical operations in MQC come about from the byproduct operators that occur during quantum teleportation. For example, given a single qubit state formula_28, one can connect this qubit to a plus state (formula_7) via a two-qubit controlled-Z operation. Then, upon measuring the first qubit (the original formula_28) in the Pauli-X basis, the original state of the first qubit is teleported to the second qubit with a measurement outcome dependent extra rotation, which one can see from the partial inner product of the measurement acting on the two-qubit state:
Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtaining—with equal probabilities—either | 0 ⟩ {\displaystyle |0\rangle } or | 1 ⟩ {\displaystyle |1\rangle } , i.e., she can not tell if her qubit has value “0” or “1”. Because of the qubits' entanglement, Bob must now get exactly the same measurement as Alice. For example, if she measures a | 0 ⟩ {\displaystyle |0\rangle } , Bob must measure the same, as | 00 ⟩ {\displaystyle |00\rangle } is the only state where Alice's qubit is a | 0 ⟩ {\displaystyle |0\rangle } . In short, for these two entangled qubits, whatever Alice measures, so would Bob, with perfect correlation, in any basis, however far apart they may be and even though both can not tell if their qubit has value “0” or “1” — a most surprising circumstance that can not be explained by classical physics.
The goal of this step is to reversibly redistribute the entropy of all qubits, such that the biases of the qubits are in descending (or non-ascending) order. The operation is done reversibly in order to prevent the entropy of the entire system from increasing (as it cannot decrease in a closed system, see entropy). In terms of temperature, this step rearranges the qubits in ascending order of temperature, so that the reset qubits are the hottest. In the example of the three qubits formula_49, this means that after the compression is done, the bias of qubit formula_51 is the highest and the bias of formula_63 is the lowest. In addition, the compression is used for the cooling of the computational qubits.
Let formula_7 be the quantum state to be transmitted. With no error correcting protocol in place, the transmitted state will be correctly transmitted with probability formula_8. We can however improve on this number by "encoding" the state into a greater number of qubits, in such a way that errors in the corresponding "logical qubits" can be detected and corrected. In the case of the simple three-qubit repetition code, the encoding consists in the mappings formula_9 and formula_10. The input state formula_1 is encoded into the state formula_12. This mapping can be realized for example using two CNOT gates, entangling the system with two ancillary qubits initialized in the state formula_13. The encoded state formula_14 is what is now passed through the noisy channel.
Quantum logic gates, building blocks for a quantum circuit in a quantum computer, operate on one, two, or three qubits: mathematically, the qubits undergo a (reversible) unitary transformation under the quantum gate. For a single qubit, unitary transformations correspond to rotations of the qubit (unit) vector on the Bloch sphere to specific superpositions. For two qubits, the Controlled NOT gate can be used to entangle or disentangle them. Standard basis measurement is an irreversible operation in which information is gained about the state of a single qubit (and coherence is lost). The result of the measurement will be either | 0 ⟩ {\displaystyle |0\rangle } (with probability | α | 2 {\displaystyle |\alpha |^{2}} ) or | 1 ⟩ {\displaystyle |1\rangle } (with probability | β | 2 {\displaystyle |\beta |^{2}} ). Measurement of the state of the qubit alters the magnitudes of α and β. For instance, if the result of the measurement is | 1 ⟩ {\displaystyle |1\rangle } , α is changed to 0 and β is changed to the phase factor e i ϕ {\displaystyle e^{i\phi }} no longer experimentally accessible. When a qubit is measured, the superposition state collapses to a basis state (up to a phase) and the relative phase is rendered inaccessible (i.e., coherence is lost). Note that a measurement of a qubit state that is entangled with another quantum system transforms the qubit state, a pure state, into a mixed state (an incoherent mixture of pure states) as the relative phase of the qubit state is rendered inaccessible.
Now we perform a "Gram-Schmidt
orthogonalization" with respect to the symplectic product.
Add row one to any other row that has one as the leftmost entry in its formula_56
matrix. Add row two to any other row that has one as the leftmost entry in its
formula_57 matrix. For our example, we add row one to row four and we add row two to
rows three and four. The matrix becomes
The first two rows are now symplectically orthogonal to all other rows per the
fundamental theorem of symplectic geometry.
We proceed with the same algorithm on the next two rows. The next two rows are
symplectically orthogonal to each other so we can deal with them individually.
Perform a Hadamard on qubit two. The matrix becomes
Perform a CNOT from qubit two to qubit three and from qubit two to qubit
four. The matrix becomes
Perform a phase gate on qubit two:
Perform a Hadamard on qubit three followed by a CNOT from qubit two to qubit
three:
Add row three to row four and perform a Hadamard on qubit two:
Perform a Hadamard on qubit four followed by a CNOT from qubit three to qubit
four. End by performing a Hadamard on qubit three:
The above matrix now corresponds to the canonical Pauli operators. Adding one half of an ebit to the receiver's side
gives the canonical stabilizer whose
simultaneous +1-eigenstate is the above state.
The above operations in reverse order
take the canonical stabilizer to the encoded
stabilizer.
In quantum computing, and more specifically in superconducting quantum computing, the phase qubit is a superconducting device based on the superconductor-insulator-superconductor (SIS) Josephson junction, designed to operate as a quantum bit, or qubit. The phase qubit is closely related, yet distinct from, the flux qubit and the charge qubit, which are also quantum bits implemented by superconducting devices.
The algorithm operates on an array of equally (and independently) biased qubits. After the algorithm transfers heat (and entropy) from some qubits to the others, the resulting qubits are rearranged in increasing order of bias. Then this array is divided into two sub-arrays: "cold" qubits (with bias exceeding a certain threshold chosen by the user) and "hot" qubits (with bias lower than that threshold). Only the "cold" qubits are used for further quantum computation. The basic procedure is called "Basic Compression Subroutine" or "3 Bit Compression".