Understanding Quantum Computing: Lecture 3 Visualizing Qubits

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Quantum Computing: Quantum Computing: Lecture 3 Lecture 3 Samuel J. Lomonaco, Jr. Dept. of Computer Science & Electrical Engineering University of Maryland Baltimore County Baltimore, MD 21250 Email: Lomonaco@UMBC.EDU WebPage: http://www.csee.umbc.edu/~lomonaco Transcribed into PowerPoint by Mark Transcribed into PowerPoint by Mark Laczin Laczin

Lecture 3: Part 1 Lecture 3: Part 1 Visualizing Qubits: Visualizing Qubits: The Bloch Sphere The Bloch Sphere

Visualizing Qubits Visualizing Qubits It s easy to visualize the state It s easy to visualize the state space space ?,? of a of a Shannon bit, i.e., Shannon bit, i.e., But how do we visualize the state of a qubit? But how do we visualize the state of a qubit? So far, we ve thought of a single qubit as a vector in a So far, we ve thought of a single qubit as a vector in a ?- -D complex Hilbert space. complex Hilbert space. But this is actually not fully correct! But this is actually not fully correct! D

Visualizing Qubits Visualizing Qubits Let Let ? ?. . Actually, for all Actually, for all ? ? , ? ??? ? ? represent the same physical state, because a global phase state, because a global phase ? is not physically detectable. Hence, is not physically detectable. Hence, ? ? and and ? represent the same state, i.e., represent the same state, i.e., ? ? ? . represent the same physical More precisely, the complex line More precisely, the complex line through the origin of state, i.e., state, i.e., through the origin of ? is actually the is actually the ? ? ? is the state. is the state.

Visualizing Qubits Visualizing Qubits The spac The space of all complex lines through the origin is called e of all complex lines through the origin is called the the complex projective space complex projective space and is denoted and is denoted ??. We now show how to visualize the qubit state space We now show how to visualize the qubit state space ??as the surface of a ball, i.e., a as the surface of a ball, i.e., a ?- -sphere. sphere.

Visualizing Qubits Visualizing Qubits Consider an arbitrarily chosen state of a qubit: Consider an arbitrarily chosen state of a qubit: ? ? ? = ? ? + ? ? = For the time being, assume that the amplitude For the time being, assume that the amplitude ? ?. Then the state is equivalent to is equivalent to Then the state ? ?? = ? +? ?? , ? ?= ? + ?? is a complex number. where where is a complex number.

Visualizing Qubits Visualizing Qubits In this way, the states of a quantum system correspond to the In this way, the states of a quantum system correspond to the complex numbers complex numbers : : ? ? ? ? . If If ? = ?, we say that the state corresponds to a point at infinity , we say that the state corresponds to a point at infinity . . So, the state space of a qubit can be identified with the complex So, the state space of a qubit can be identified with the complex numbers numbers plus infinity plus infinity , i.e., , i.e., .

Visualizing Qubits: Visualizing Qubits: The Complex Plane The Complex Plane ?= ? ? ? ??= ? ? + ? ? + ? ? + = +? ? = ? ? ? ? ? ? ? ? = ? ? = ? ? ? ??= ? + ?? ? = ? ? + ? ?

Visualizing Qubits: Visualizing Qubits: The Complex Plane The Complex Plane ? ?= ? + ?? |? > +? | + ? > | +> | > ? ?|? > ? ? ? | ? >

VISUALIZING QUBITS: VISUALIZING QUBITS: THE BLOCH SPHERE THE BLOCH SPHERE |? > We now use the We now use the reverse stereographic projection stereographic projection to identify to identify with the surface of a ball, the surface of a ball, i.e., the i.e., the ?- -sphere. sphere. reverse |? > |? > with ?/? ? | > ? ? ? | +> | ? > |? >

Visualizing Qubits: Visualizing Qubits: The Bloch Sphere The Bloch Sphere Thus, the state space of a qubit is a sphere, called the Thus, the state space of a qubit is a sphere, called the Bloch sphere Bloch sphere. .

Lecture 3: Part 2 Lecture 3: Part 2 Multipartite Systems: Multipartite Systems: Beyond a Single Qubit Beyond a Single Qubit

Definitions A multipartite quantum system ? consists of ? qubits ??,??, ,?? ?, with respective Hilbert spaces ??,??, ,?? ?, and has as its state space the Hilbert space ? ???. ? = ?=?

Definitions If ? , ? is designated as the standard orthonormal basis of ??, then these bases induce a standard orthonormal basis ? , ? , , ?? ? of the ? qubit state space ?,where the above integer labels denote the corresponding n n bit binary strings.

The State of ? Thus, the state ? of ? qubits is of the form ?? ? ??? = ??? + ??? + + ??? ??? ? ? = ?=?

Shannon Bits Note that it is easy to visualize the state space of ? Shannon bits. This is because the state space of Shannon bits, ?,??= ?,? ?,? ?,? , is simply the set of all binary ? bit strings.

Bits versus Qubits For classical bits, please note that the number of states of an ? bit multipartite system is ? ??, but each state can be easily designated by an ? ? -bit string. For qubits, the state space ? of a multipartite quantum system is much more complex, and more difficult to visualize.

Question: How to visualize an ?-qubit state? ?? ???? be a state of ? qubits. Let ? = ?=? Case 1: if ?? ?, then this is the same physical state as ?? ??? ?? ?? ??,?? ??, ,??? ? ?? ? ? = ? + ? ?? ?=? Case 2: iI ??= ? and ?? ?, then ?? ??? ?? ??,?? ??, ,??? ? ?? ? ? = ? + ? ?? ?? ?=? Case 3: etc

Question continued It follows that the state space of ? qubits can be identified with ?? ? ?? ? , which is a ?? ? -dimensional complex space, which in turn is a 2 ?? ? -dimensional Real space. This space is the complex projective space ??? ?.

Lecture 3: Part 3 Lecture 3: Part 3 Measurement: Measurement: Classical and Quantum Classical and Quantum

Measurement: W.r.t the Measurement: W.r.t the ? , ? basis basis Recall that if a qubit in the state Recall that if a qubit in the state ? = ? ? + ? ? is measured with respect to the standard basis measured with respect to the standard basis ? , ? , then then Measurement Measurement w.r.t. w.r.t. ? , ? ? ? , ??= ?? ? , ??= ?? where we have assumed that where we have assumed that ? is of unit length, i.e., is of unit length, i.e., ? ? = ??+ ??= ?.

Measurement: Measurement: W.r.t. W.r.t. other bases, e.g., other bases, e.g., + , Now we also note that measurement may be made with Now we also note that measurement may be made with respect to any orthonormal basis. respect to any orthonormal basis. For example, let us consider the For example, let us consider the orthnormal ? + ? orthnormal basis ? ? basis + = = ? ? If we re If we re- -express express ? in this basis, we have in this basis, we have ? + ? ? ? ? ? ? = + + Then, Then, ? ?+? ? Measurement Measurement + , ?+= ? w.r.t. w.r.t. + , ? ? ? ? , ? =

Measurement: Measurement: W.r.t. In like manner, consider the In like manner, consider the orthnormal W.r.t. the the ? , ? basis orthnormal basis basis basis ? + ? ? ? ? ? ? = ? = ? ? If we re If we re- -express express ? in this basis, we have in this basis, we have ? ?? ? ? + ?? ? ? = ? + ? Then, Then, ? ? ?? ? ? , ??= Measurement Measurement ? w.r.t. w.r.t. ? ? , ? ?+?? ? ? , ? ?=

Classical versus Quantum Measurement: As coin flips The previous three examples of measurements w.r.t. a basis demonstrate very emphatically how a qubit differs from a Shannon bit. We can see this with an example. Consider a classical coin, which when flipped (thus observed) produces a heads H H or a tails T T with probability ?, ?, where ??+ ??= ?. ?? ??

Measurement: Qubits Let us now compare the classical coin flip with the measurement of a qubit in the state ? = ? ? + ? ? , where ??+ ??= ?. Then, unlike a classical coin, the qubit behaves as 3 different coins, depending on which of the three bases ? , ? , + , , ? , ? is chosen, as shown in the following figure:

Measurement of a qubits ? Qubit Qubitt Measure Measure ? , ? ??= ?? ??= ?? ? , ? ,

Measurement: more qubits In other words, a qubit behaves like many different coins, depending on how it is observed. If, in the same vein, we were to continue to pursue this for 2 qubits, we would encounter some amazing non- classical behavior, e.g., the Bell inequalities.

There is much more to Q Measurement There is much more to Q Measurement Step 1. Measurement of a qubit with respect to orthonormal basis Step 2. Measurement of n qubits with respect to orthonormal basis Step 3. Measurement of with respect to eigen basis of a Hermitian operator (a.k.a., an observable

Lecture 3: Part 4 Lecture 3: Part 4 Wiring Diagrams: Wiring Diagrams: Drawing Quantum Circuits Drawing Quantum Circuits

Wiring Diagrams: Wiring Diagrams: Recall Recall As previously mentioned, there are two types of quantum As previously mentioned, there are two types of quantum computer instruction: computer instruction: 1. 1. Unitary transformations Unitary transformations ?, where 2. 2. Measurements, Measurements, which are given by Hermitian matrices which are given by Hermitian matrices ?, where where ? = ?. However, we now encounter a problem However, we now encounter a problem where ? = ? ?.

A Problem A Problem As the number As the number ? of qubits grows, the size of the matrices of qubits grows, the size of the matrices ? and and ? grow as grow as ?? ??, i.e., i.e., exponentially! Question: How do we program a quantum computer when Question: How do we program a quantum computer when the size of the instructions grows as the size of the instructions grows as ? ??? ? The answer is to represent quantum computer programs The answer is to represent quantum computer programs as as wiring diagrams wiring diagrams which grow linearly which grow linearly ? ? with In a sense, the language of In a sense, the language of wiring diagrams wiring diagrams is analogous to today s classical computer languages, e.g., C/C++, to today s classical computer languages, e.g., C/C++, Python, etc. Python, etc. exponentially! with ?. is analogous

What is a wiring diagram? What is a wiring diagram? A wiring diagram consists of A wiring diagram consists of lines A solid line corresponds to a qubit, and a double A solid line corresponds to a qubit, and a double line corresponds to a classical Shannon bit. line corresponds to a classical Shannon bit. A solid line labeled with a slash denotes A solid line labeled with a slash denotes ? qubits. Gates correspond to quantum computer instructions, i.e., Gates correspond to quantum computer instructions, i.e., to unitary transformations and measurements. to unitary transformations and measurements. lines and and gates gates. . qubits. ?

Some single qubit gates Some single qubit gates ? ? ? ? Hadamard: Hadamard: = ? ? ? ? ? ? Pauli Pauli- -X X: : = ? ? ? ? Pauli Pauli- -Y Y: : =

Some single qubit gates ? ? ? Pauli Pauli- -Z Z: : = ? ? ? ? ? Phase: Phase: = ? ? ? ? ?:= ???/?

Some multi Some multi- -qubit gates pt. 1 qubit gates pt. 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Controlled Controlled- -NOT: NOT: = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? SWAP: = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Controlled Controlled- -Z Z: = ?

Some multi Some multi- -qubit gates qubit gates Toffoli: Fredkin:

Measurement gate Measurement gate Measurement Measurement is represented by a small meter: is represented by a small meter:

Composition Composition In general, a wiring diagram of the form In general, a wiring diagram of the form And represents the unitary transformation And represents the unitary transformation ?? ??? ? ????. Note the corresponding Note the corresponding order reversal order reversal of the product! of the product!

Example 1 Example 1 Consider the following wiring diagram: Consider the following wiring diagram: This is equivalent to This is equivalent to ???? ? ? ? ? That is, That is, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ?

Example 1 cont. Performing the arithmetic, Performing the arithmetic, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ?? + ?? = + = ? ? ? which is which is known as a known as a Bell state Bell state.

Exercise 1 Show that Show that =

Exercise 2 Show that Show that = = Note that in general, Note that in general, ? ? ? ? = ? ? ? ? = ? ?

Now consider Now consider ? ?? , ???= ?? , ???= ? ?? , ???= ? ?? , ???= ? = ? ?

The Bell Orthonormal Basis The Bell Orthonormal Basis Consider the Consider the Bell Bell orthnormal orthnormal basis basis of a of a ?- -qubit state space: qubit state space: ?? + ?? ??? = ? ? ??? = ? ?? + ?? ??? = ? ? ??? = ? ?? ?? ??? = ? ? ??? = ? ?? ?? ??? = ?? ? ??? = ? Thus, Thus, ??? = ???? ? ???where where ?,? ?,? .

Note Note We can move from one Bell basis to any other by We can move from one Bell basis to any other by applying a local unitary transformation to only the left applying a local unitary transformation to only the left qubit. qubit. Later on, this will be useful in quantum communication Later on, this will be useful in quantum communication protocols. protocols. Exercise 3: Verify the formulas relating the Bell basis Exercise 3: Verify the formulas relating the Bell basis elements. elements.

Transforming the Bell Basis Transforming the Bell Basis We can translate the equation We can translate the equation ??? = ???? ? ??? Into a wiring diagram as follows: Into a wiring diagram as follows: where gates where gates ??and classical bits classical bits ?,?. In other words, and ?? are gates controlled by external are gates controlled by external . In other words, ??= ?, ?? ? = ? ?, ?? ? = ? and Please note that the first two gates from the left produce Please note that the first two gates from the left produce the Bell state the Bell state ??? . . and ??= ?, ?? ? = ? ?, ?? ? = ?

Exercise 4 Exercise 4 Verify that Verify that ??? Thus, the above unitary operations change the Thus, the above unitary operations change the standard basis into the Bell basis basis into the Bell basis. . standard

Exercise 5 Exercise 5 Prove that Prove that Thus, the above unitary operation transforms the Thus, the above unitary operation transforms the Bell basis into the standard basis basis into the standard basis. . Bell