Quantum circuit
The minimum set of actions that a circuit needs to be able to perform on the qubits to enable quantum computation is known as DiVincenzo's criteria.
Circuits are written such that the horizontal axis is time, starting at the left hand side and ending at the right.
[citation needed] Richard Feynman used an early version of the quantum circuit notation in 1986.
This space can also be regarded as consisting of linear combinations, or superpositions, of classical bit strings.
However, the Hilbert-space structure of the qubits permits many quantum gates that are not induced by classical ones.
To explain this assembly process, suppose we have a reversible n-bit gate f and a reversible m-bit gate g. Putting them together means producing a new circuit by connecting some set of k outputs of f to some set of k inputs of g as in the figure below.
We will refer to this scheme as a classical assemblage (This concept corresponds to a technical definition in Kitaev's pioneering paper cited below).
This condition assures that intermediate "garbage" is not created (the net physical effect would be to increase entropy, which is one of the motivations for going through this exercise).
This means that given any reversible classical n-bit circuit h, we can construct a classical assemblage of Toffoli gates in the above manner to produce an (n+m)-bit circuit f such that where there are m underbraced zeroed inputs and Notice that the result always has a string of m zeros as the ancilla bits.
Obviously, if the mapping fails to be injective, at some point in the simulation (for example as the last step) some "garbage" has to be produced.
In a real quantum computer the physical connection between the gates is a major engineering challenge, since it is one of the places where decoherence may occur.
We now provide a mathematical model for how quantum circuits can simulate probabilistic but classical computations.
Let us assume therefore that the initialization is a mixed state given by some density operator S which is near the idealized input in some appropriate metric, e.g.
Note that observables in quantum mechanics are usually defined in terms of projection valued measures on R; if the variable happens to be discrete, the projection valued measure reduces to a family {Eλ} indexed on some parameter λ ranging over a countable set.
Similarly, a Y valued observable, can be associated with a family of pairwise orthogonal projections {Ey} indexed by elements of Y. such that Given a mixed state S, there corresponds a probability measure on Y given by The function F:X → Y is computed by a circuit U:HQB(r) → HQB(r) to within ε if and only if for all bitstrings x of length m Now so that Theorem.
With the advent of quantum computing, there has been a significant surge in both the number of developers and available tools.
[7] However, the slow pace of technological advancement and the high maintenance costs associated with quantum computers have limited broader participation in this field.
The fundamental advantage of quantum computers lies in their ability to process qubits, leveraging properties like entanglement and superposition simultaneously.
FPGA is a kind of hardware that excels at executing operations in parallel, supports pipelining, has on-chip memory resources with low access latency, and offers the flexibility to reconfigure the hardware architecture on-the-fly which make it a well suited tool to handle matrix multiplication.
Suppose we are simulating 5-qubit circuits, then we need to store the vector that holds 32 (2⁵) 16-bit values, each of which represents the square-root probability of a possible existing state.
This finding underscores the feasibility of leveraging FPGAs to accelerate quantum computing simulations.
