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Mathematically qubits are represented by column vectors and the logical operations to manipulate them (called gates) are represented by matrices. Consider the 0-bit and 1-bit, 10) = (6) |1) = (4) These form the basis of a single qubit |p) which is generally a complex linear combination of the two bits, |p) = co |0) + c1 [1), with co,ci € C. Consider the simplest possible case where the qubit is prepared in either one of the basis vectors, namely |4) = |0) or |p) = |1)

Mathematically qubits are represented by column vectors and the logical operations to manipulate them (called gates) are represented by matrices. Consider the 0-bit and 1-bit, 10) = (6) |1) = (4) These form the basis of a single qubit |p) which is generally a complex linear combination of the two bits, |p) = co |0) + c1 [1), with co,ci € C. Consider the simplest possible case where the qubit is prepared in either one of the basis vectors, namely |4) = |0) or |p) = |1)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Mathematically qubits are represented by column vectors and the logical operations to manipulate them (called gates) are represented by matrices. Consider the 0-bit and 1-bit, |0) = (6) |1) = (4) These form the basis of a single qubit |p) which is generally a complex linear combination of the two bits, |p) = co |0) +c1 |1), with co,c1 € C. Consider the simplest possible case where the qubit is prepared in either one of the basis vectors, namely |Þ) = |0) or |4) = |1) (a) Compute the action of the X-gate on |0). Do this though matrix X |0) where, (: ) X = (b) Repeat your calculation in (a), but this time calculating X|1). X is called the NOT-gate. Explain this in terms of your results from (a)-(b). It will help to think of |0) corresponding to FALSE, |1) corresponding to TRUE, and what the action of the matrix does.
Transcribed Image Text:Mathematically qubits are represented by column vectors and the logical operations to manipulate them (called gates) are represented by matrices. Consider the 0-bit and 1-bit, |0) = (6) |1) = (4) These form the basis of a single qubit |p) which is generally a complex linear combination of the two bits, |p) = co |0) +c1 |1), with co,c1 € C. Consider the simplest possible case where the qubit is prepared in either one of the basis vectors, namely |Þ) = |0) or |4) = |1) (a) Compute the action of the X-gate on |0). Do this though matrix X |0) where, (: ) X = (b) Repeat your calculation in (a), but this time calculating X|1). X is called the NOT-gate. Explain this in terms of your results from (a)-(b). It will help to think of |0) corresponding to FALSE, |1) corresponding to TRUE, and what the action of the matrix does.
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