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Two qubit Gates, GHZ States & Entanglement | Dr. Stefan Seegerer
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Overview
This video explains the fundamental concepts of multi-qubit systems in quantum computing, focusing on two-qubit gates, entanglement, and the GHZ state. It details how adding qubits exponentially increases computational space, making quantum computers vastly more powerful than classical ones for certain problems. The session covers the mathematical representation of quantum states, the functionality of key two-qubit gates like CNOT, and demonstrates how these gates can create entangled states. It also touches upon the practical aspects of running quantum circuits on real hardware, including transpilation and routing, and introduces methods for verifying entanglement.
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Chapters
- Quantum computers gain significant power by using multiple qubits, as the computational space grows exponentially with each added qubit (2^n states for n qubits).
- A single qubit can be in a superposition of two states (0 and 1), while two qubits can be in a superposition of four states (00, 01, 10, 11) simultaneously.
- This exponential growth in computational space allows quantum computers to explore many possibilities at once, enabling solutions to problems intractable for classical computers.
- Even with 50 qubits, the number of states (2^50) exceeds the capacity of the largest supercomputers, highlighting the unique advantage of quantum hardware.
Understanding the exponential growth of computational space with more qubits is crucial for appreciating why quantum computers have the potential to solve complex problems that are impossible for even the most powerful classical supercomputers.
A system of 50 qubits can represent approximately a quadrillion (2^50) basis states simultaneously, a scale that classical computers cannot manage.
- A single qubit state is mathematically represented as a linear combination of basis states (ket 0 and ket 1) with complex coefficients, normalized so the sum of squared magnitudes is one.
- These states can be visualized as vectors, and single-qubit operations (gates) are represented by matrices that act on these vectors through multiplication.
- For multi-qubit systems, states are described by combining individual qubit states, often using the tensor product, leading to larger vectors and matrices.
- Two-qubit systems require 4-component vectors and 4x4 matrices to describe their states and operations, respectively.
Learning how quantum states and gates are mathematically represented is essential for understanding how quantum computations are performed and for designing quantum algorithms.
The Hadamard gate is an example of a single-qubit gate that transforms a basis state (like ket 0) into a superposition state (like ket plus).
- The CNOT (Controlled-NOT) gate is a fundamental two-qubit gate where the state of a second (target) qubit is flipped if and only if the first (control) qubit is in the state '1'.
- When the control qubit is in a superposition, the CNOT gate can create entangled states.
- Entanglement is a quantum phenomenon where two or more qubits become linked, sharing a single quantum state such that measuring one instantly influences the state of the others, regardless of distance.
- Entangled states, like the Bell states, cannot be described as simple products of individual qubit states; the system must be considered as a whole.
The CNOT gate is a building block for quantum computation, and its ability to create entanglement is key to unlocking the power of quantum algorithms.
Applying a Hadamard gate to the first qubit, followed by a CNOT gate with the first qubit as control and the second as target, creates a Bell state (1/sqrt(2) * |00> + 1/sqrt(2) * |11>), where measuring either qubit instantly reveals the state of the other.
- A universal set of quantum gates, typically consisting of single-qubit gates and one type of two-qubit gate (like CNOT), is sufficient to perform any quantum computation.
- While multi-qubit gates exist (e.g., Toffoli), any N-qubit unitary gate can be decomposed into single-qubit gates and CNOT gates.
- The GHZ (Greenberger-Horne-Zeilinger) state is a generalization of Bell states to multiple qubits, characterized by having only the all-zero and all-one components.
- GHZ states are highly entangled and can be created using a sequence of Hadamard and CNOT gates, demonstrating entanglement across more than two qubits.
Understanding that a limited set of gates is universal simplifies the design of quantum computers and algorithms, while GHZ states showcase advanced multi-qubit entanglement.
A GHZ state for three qubits can be represented as (1/sqrt(2) * |000> + 1/sqrt(2) * |111>), where all qubits are either 0 or all are 1 upon measurement.
- Quantum circuits designed on abstract models must be translated (transpiled) into a sequence of native gates supported by specific quantum hardware.
- Hardware connectivity (topology) dictates how qubits can interact, often requiring swap gates to move information between non-adjacent qubits, which adds overhead and potential errors.
- Optimizing circuit layout to match hardware topology and minimize gate operations is crucial for performance and reducing noise.
- Real quantum hardware is subject to noise, which can introduce errors and deviate results from ideal predictions, making verification essential.
Bridging the gap between theoretical quantum circuits and their execution on physical devices involves complex processes like transpilation and routing, which are critical for practical quantum computing.
A CNOT gate might be decomposed into CSET gates and single-qubit rotations on IQM hardware, and a required operation between distant qubits might be implemented using multiple swap gates.
- Entanglement can be verified by measuring the 'state fidelity,' comparing the experimentally obtained state to an ideal state.
- A fidelity above 0.5 for a GHZ state indicates that the results cannot be explained without entanglement involving all qubits, confirming 'quantumness'.
- Quantum state tomography is a method to fully reconstruct the quantum state (density matrix) but is resource-intensive for larger systems.
- The method of 'multiple quantum coherences' offers a more efficient way to measure off-diagonal elements of the density matrix, enabling entanglement verification with fewer measurements.
Verifying that entanglement has been successfully created and that the system exhibits genuine quantum behavior is essential for validating quantum computations and hardware.
Measuring a GHZ state and calculating its fidelity; if the fidelity is significantly above 0.5, it proves that the system is genuinely entangled across all involved qubits.
Key takeaways
- The exponential scaling of computational space with the number of qubits is the primary source of quantum computing's potential power.
- Entanglement is a non-classical correlation between qubits that is essential for many quantum algorithms and cannot be simulated efficiently by classical computers.
- Two-qubit gates, such as CNOT, are sufficient alongside single-qubit gates to perform any quantum computation (universal gate set).
- The GHZ state is a key example of multi-qubit entanglement, demonstrating correlations beyond pairwise entanglement.
- Executing quantum algorithms on real hardware requires transpilation to native gates and careful consideration of qubit connectivity and noise.
- Verifying entanglement and quantumness is a critical step in validating quantum hardware and experiments, often using metrics like state fidelity.
Key terms
QubitSuperpositionEntanglementKet notationDirac notationTwo-qubit gateCNOT gateBell stateGHZ stateTranspilationState fidelityQuantum state tomographyMultiple quantum coherences
Test your understanding
- How does the computational space of a quantum computer scale with the number of qubits, and why is this exponential growth significant?
- What is entanglement, and how does it differ from classical correlations between systems?
- Explain the function of the CNOT gate and how it can be used to create entangled states.
- Why are two-qubit gates considered universal for quantum computation when combined with single-qubit gates?
- What are the main challenges when running a quantum circuit on real quantum hardware, and how are they addressed?