In quantum computing, quantum parallelism is the capacity of a quantum computer to assess a function on several inputs at the same time due to the superposition principle, therefore enabling simultaneous processing across superposed states, which is essential for the speedup observed in quantum algorithms.
Superposition
Existing in a superposition of states, quantum bits (qubits) can simultaneously represent many values. With respective probability specified by the coefficients of the state vector, a single qubit can, for instance, be in a state that simultaneously represents ∣0⟩ and ∣1⟩.
Parallel computation
Applied to a qubit in superposition, a quantum operation—or gate—operates on all components of the superposition simultaneously. This enables a quantum computer to do concurrent operations devoid of extra resources. For a quantum system in a superposition of n states, for instance, a single operation can concurrently calculate the outcome for all n states.
Limitations
Quantum parallelism does not equal instantaneous access to all parallel results. Measuring a quantum system reveals it to be collapsed to a single outcome, therefore losing knowledge about alternative states. Extensive relevant information must be effectively extracted from the quantum parallelism via creative algorithm design.
An example of quantum parallelism
Classic example of quantum parallelism is the Deutsch-Jozsa algorithm. This approach uses less evaluations than traditional techniques to ascertain if a given function is constant or balanced. Leveraging superposition, the method concurrently evaluates the function on every conceivable input.
- Initialize: Beginning with two qubits in ∣0⟩∣1⟩,
- Superposition: Applying a Hadamard gate to the first qubit generates 1/√2(∣0⟩+∣1⟩).
- Function Evaluation: Entangle the superposition’s function f(x) with a unitary Uf.
- Interfere: Apply another Hadamard gate to extract globally either constant or balanced f(x).
- Measure: the first qubit to expose the outcome.
How Quantum Parallelism works, step-by-step?
By constructing a superposition of inputs and developing it with a quantum algorithm, quantum parallelism lets a quantum computer assess a function for several inputs concurrently. Combining this capacity with the quantum events of entanglement and interference helps quantum computers to solve insurmountable issues for conventional computers.
Quantum Representation
Quantum bits (qubits) can exist in a superposition of both states unlike classical bits that are 0 or 1. α|0⟩ + β|1⟩ defines the state of a qubit via a linear combination of the base states |0⟩ and |1⟩. Here, α and β are complex integers in which the square of their magnitudes determines either the chance of measuring |0⟩ or |1⟩ correspondingly. A qubit then lives in a mix of values rather than only one value.
Multi-Qubits Superposition
Combining many qubits creates a quantum record that can be found in a superposition of all conceivable combinations of the individual qubit states. An n-qubit register can concurrently be in a superposition of 2n states. A two-qubit system can be in a superposition of |00⟩, |01⟩, |10⟩, and |11⟩, for instance.
Quantum Operation
Quantum gates—unitary transformations working on the qubits—are used in quantum algorithms. Applied to a qubit or a qubit register in superposition, a quantum gate acts on all the states in the superposition concurrently. This implies that for any x, you obtain a superposition of the outputs f(x) by applying a quantum operator Uf to a superposition of inputs. The quantum gates control the multifarious amplitudes connected with every state in the superposition. This is somewhat distinct from classical operations acting on one state at a time.
Uf=1/√2 ∣f(x)⟩ ∣x⟩
Every ∣x⟩ is matched with its corresponding ∣f(x)⟩, therefore parallel computation of f(x) is accomplished for every x.
Parallel Computing
Quantum parallelism is the simultaneous action on all feasible states. One operation on n qubits may essentially simultaneously do calculations on 2n distinct inputs. A fundamental aspect of quantum computing is this exponential scaling of the computational space with the qubit count.
Interference and Measurement
While measurement collapses the superposition to a single conventional conclusion, quantum parallelism lets several calculations occur simultaneously. The key is to control the quantum states such that the desirable outcomes interact constructively, raising their probability, and the unwanted outcomes interact destructively, therefore lowering their probability. This means, measured, the right or intended response is far more likely.
What is the difference between classical parallelism and quantum parallelism?
| Feature | Classical Parallelism | Quantum Parallelism |
|---|---|---|
| Underlying Principles | Classical physics | Quantum mechanics |
| Information Storage | Bits (0 or 1) using voltage or charge | Qubits (superposition of 0 and 1) often using electron spin |
| Parallelism Method | Multiple processors working simultaneously on different tasks | Superposition, enabling a single quantum computer to exist in multiple states concurrently |
| Superposition | No concept of superposition | Qubits exist in a combination of states |
| Operations | Based on Boolean algebra and have local effects | Based on linear algebra over Hilbert space, potentially having non-local effects due to entanglement |
| Computational Space | Increases linearly with the number of processors | Increases exponentially with the number of qubits due to superposition. An n-qubit system can be in a superposition of 2n states. |
| Execution | Multiple processors execute different parts of the problem | Operations performed on all states in superposition simultaneously |
| Interference | No concept of interference | Quantum algorithms exploit interference to amplify desired outcomes and suppress undesired ones |
| Measurement | No concept of measurement | Measurement collapses the superposition to a single classical outcome |
| Speedup | Linear speedup based on number of processors | Potential for exponential speedup over classical algorithms for certain problems due to superposition, interference, and entanglement |
| Applications | General-purpose parallel processing | Crucial for enabling quantum algorithms to tackle problems that are intractable for classical computers, such as factorization (Shor’s algorithm) and database searching (Grover’s algorithm) |