Outline Quantum Bits So What? Other Topics and Open Problems Quantum Computing - An Introduction Cris Cecka April 10, 2006 Cris Cecka Quantum Computing - An Introduction
Outline Quantum Bits So What? Other Topics and Open Problems Quantum Bits Quantum Superposition Dirac Properties Orthogonality and Bases Operators So What? Interesting Operators Quantum Computing Strategies Amplitude Amplification - A Start Grover’s Algorithm Other Topics and Open Problems Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators (Qu)Bits Classical Bits (Bits) Quantum Bits (Qubits) ◮ Two States: ◮ Two States: Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators (Qu)Bits Classical Bits (Bits) Quantum Bits (Qubits) ◮ Two States: ◮ Two States: ◮ Spin up, Energy State ◮ On, Up, High Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators (Qu)Bits Classical Bits (Bits) Quantum Bits (Qubits) ◮ Two States: ◮ Two States: ◮ Spin up, Energy State ◮ On, Up, High ⇒ | 1 � ⇒ 1 Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators (Qu)Bits Classical Bits (Bits) Quantum Bits (Qubits) ◮ Two States: ◮ Two States: ◮ Spin up, Energy State ◮ On, Up, High ⇒ | 1 � ⇒ 1 ◮ Spin down, Energy State ◮ Off, Down, Low Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators (Qu)Bits Classical Bits (Bits) Quantum Bits (Qubits) ◮ Two States: ◮ Two States: ◮ Spin up, Energy State ◮ On, Up, High ⇒ | 1 � ⇒ 1 ◮ Spin down, Energy State ◮ Off, Down, Low ⇒ | 0 � ⇒ 0 Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators At Least It Sounds Good Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators Quantum Superposition Definition Qubits can be BOTH | 0 � and | 1 � . In general, a single qubit is � α � | Ψ � = α | 0 � + β | 1 � = β where α 2 + β 2 = 1. Problem This superposition only occurs when we aren’t “looking”. When we “look” ◮ Measure | 0 � with probability α 2 . ◮ Measure | 1 � with probability β 2 . Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators Speaking The Same Language Definition � a � Kets: | x � = a | 0 � + b | 1 � = b Can be thought of as vectors. Definition � a � ∗ � b ∗ � Bras: � x | = a ∗ � 0 | + b ∗ � 1 | = a ∗ = = | x � ∗ b Can be thought of as vectors. The ultimate fate of a bra is to meet a ket b ∗ � � a � = a 2 + b 2 � � x | x � = a ∗ b Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators Orthogonality and Bases We’ve been using {| 0 � , | 1 �} as a basis. � 1 � 0 � � Note: | 0 � = and | 1 � = are orthogonal: 0 1 � 0 | 1 � = � 1 | 0 � = 0 � 0 | 0 � = � 1 | 1 � = 1 Cris Cecka Quantum Computing - An Introduction
Outline Quantum Superposition Quantum Bits Dirac Properties So What? Orthogonality and Bases Other Topics and Open Problems Operators Operators The identity operator � 1 � 0 ˆ I = 0 1 can be written ˆ I = | 0 �� 0 | + | 1 �� 1 | Example ˆ I | Ψ � = ( | 0 �� 0 | + | 1 �� 1 | ) ( α | 0 � + β | 1 � ) = α | 0 �� 0 | 0 � + β | 0 �� 0 | 1 � + α | 1 �� 1 | 0 � + β | 1 �� 1 | 1 � = α | 0 � + β | 1 � Cris Cecka Quantum Computing - An Introduction
Outline Interesting Operators Quantum Bits Quantum Computing Strategies So What? Amplitude Amplification - A Start Other Topics and Open Problems Grover’s Algorithm Consider... Definition (Hadamard Transform) � � 1 1 ˆ 1 1 H = 2 ( | 0 �� 0 | + | 1 �� 0 | − | 0 �� 1 | + | 1 �� 1 | ) = √ √ − 1 1 2 Also called the Mixing Operator. For example, 1 | 0 � | 1 � ✔ ✕ ✔ ✕ 1 1 H | 0 � = ˆ ˆ H = √ = √ − √ 0 − 1 2 2 2 1 | 0 � | 1 � ✔ 0 ✕ ✔ 1 ✕ H | 1 � = ˆ ˆ H = √ = √ + √ 1 1 2 2 2 and again, ✒ | 0 � | 1 � 1 ✓ ✒ ✔ 1 ✕ ✔ 0 ✕✓ ˆ ˆ + ˆ H √ + √ = √ H H 0 1 2 2 2 1 ✒✔ ✕ ✔ ✕✓ 1 1 = + = | 1 � − 1 1 2 Cris Cecka Quantum Computing - An Introduction
Outline Interesting Operators Quantum Bits Quantum Computing Strategies So What? Amplitude Amplification - A Start Other Topics and Open Problems Grover’s Algorithm Quantum Computing Strategies ◮ Use superposition to parallelize computations. Cris Cecka Quantum Computing - An Introduction
Outline Interesting Operators Quantum Bits Quantum Computing Strategies So What? Amplitude Amplification - A Start Other Topics and Open Problems Grover’s Algorithm Quantum Computing Strategies ◮ Use superposition to parallelize computations. ◮ Problem is when we measure the qubit, the result is one state, not the entire superposition. ◮ Want a lot more information than is available. Cris Cecka Quantum Computing - An Introduction
Outline Interesting Operators Quantum Bits Quantum Computing Strategies So What? Amplitude Amplification - A Start Other Topics and Open Problems Grover’s Algorithm Quantum Computing Strategies ◮ Use superposition to parallelize computations. ◮ Problem is when we measure the qubit, the result is one state, not the entire superposition. ◮ Want a lot more information than is available. ◮ A good strategy: encode solutions and make wrong answers “cancel out” Cris Cecka Quantum Computing - An Introduction
Outline Interesting Operators Quantum Bits Quantum Computing Strategies So What? Amplitude Amplification - A Start Other Topics and Open Problems Grover’s Algorithm Quantum Computing Strategies ◮ Use superposition to parallelize computations. ◮ Problem is when we measure the qubit, the result is one state, not the entire superposition. ◮ Want a lot more information than is available. ◮ A good strategy: encode solutions and make wrong answers “cancel out” ◮ Problem is this is really hard. ◮ Since we can’t see the entire quantum state, most algorithms have to be oblivious. Cris Cecka Quantum Computing - An Introduction
Outline Interesting Operators Quantum Bits Quantum Computing Strategies So What? Amplitude Amplification - A Start Other Topics and Open Problems Grover’s Algorithm Quantum Computing Strategies ◮ Use superposition to parallelize computations. ◮ Problem is when we measure the qubit, the result is one state, not the entire superposition. ◮ Want a lot more information than is available. ◮ A good strategy: encode solutions and make wrong answers “cancel out” ◮ Problem is this is really hard. ◮ Since we can’t see the entire quantum state, most algorithms have to be oblivious. ◮ Amplitude amplification: amplify correct answers. Cris Cecka Quantum Computing - An Introduction
Outline Interesting Operators Quantum Bits Quantum Computing Strategies So What? Amplitude Amplification - A Start Other Topics and Open Problems Grover’s Algorithm Consider the operator H = 2 � � M = ˆ ˆ 2 | 0 �� 0 | − ˆ ˆ � | i �� j | − ˆ H I I N i , j on a general state | Ψ � = � k α k | k � . �� � 2 ˆ � | i �� j | − ˆ M | Ψ � = I α k | k � N i , j k = 2 � − � α j | i � α k | k � N i , j k 2 � j α j � � = | i � − α k | k � N i k � = (2 � α � − α k ) | k � k This operator flips all the amplitudes about their average! Cris Cecka Quantum Computing - An Introduction
Outline Interesting Operators Quantum Bits Quantum Computing Strategies So What? Amplitude Amplification - A Start Other Topics and Open Problems Grover’s Algorithm Quantum Searching ◮ Given an oracle operator ˆ O f | x � = ( − 1) f ( x ) | x � ◮ f is like a search function. Cris Cecka Quantum Computing - An Introduction
Outline Interesting Operators Quantum Bits Quantum Computing Strategies So What? Amplitude Amplification - A Start Other Topics and Open Problems Grover’s Algorithm Quantum Searching ◮ Given an oracle operator ˆ O f | x � = ( − 1) f ( x ) | x � ◮ f is like a search function. ◮ Start with | 1 �| 1 �| 1 � · · · = | 1 � ⊗ n Cris Cecka Quantum Computing - An Introduction
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