Can one use a representation that is more compact, in the sense that it uses less memory and/or computational power than the simple vector representation? How does it work?

Source: "Multiple Qubits":

"A single qubit can be trivially modeled, simulating a fifty-qubit quantum computation would arguably push the limits of existing supercomputers. Increasing the size of the computation by only one additional qubit doubles the memory required to store the state and roughly doubles the computational time. This rapid doubling of computational power is why a quantum computer with a relatively small number of qubits can far surpass the most powerful supercomputers of today, tomorrow and beyond for some computational tasks.".

So you can't utilize a Ponzi scheme or rob Peter to pay Paul. Compression will save memory at the cost of computational complexity, or representation in a more flexible space (larger) would reduce computational complexity but at a cost of memory. Essentially what is needed is more capable hardware or smarter algorithms.

As long as I don't have to explain these papers any better than they do here are some methods:

• Analogous to operand compression:

Computing depth-optimal decompositions of logical operations: "A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits" or this Quora discussion on "Encoding the dimension of the particle".

• Analogous to memory compression:

Qutrit factorization using ternary arithmetic: "Factoring with Qutrits: Shor's Algorithm on Ternary and Metaplectic Quantum Architectures" and "Quantum Ternary Circuit Synthesis Using Projection Operations".

• Analogous to traditional optimization

"A Quantum Algorithm for Finding Minimum Exclusive-Or Expressions".

• Other:

Krull Dimensions or axiomatisation and graph rewriting: "Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics".

By combining those techniques you ought to be able to squeeze the foot into the shoe. That would permit emulation of larger systems on conventional processors, just don't ask me to explain doctoral level work or write the code. :)

Can one use a representation that is more compact, in the sense that it uses less memory and/or computational power than the simple vector representation? How does it work?

Source: "Multiple Qubits":

"A single qubit can be trivially modeled, simulating a fifty-qubit quantum computation would arguably push the limits of existing supercomputers. Increasing the size of the computation by only one additional qubit doubles the memory required to store the state and roughly doubles the computational time. This rapid doubling of computational power is why a quantum computer with a relatively small number of qubits can far surpass the most powerful supercomputers of today, tomorrow and beyond for some computational tasks.".

So you can't utilize a Ponzi scheme or rob Peter to pay Paul. Compression will save memory at the cost of computational complexity, or representation in a more flexible space (larger) would reduce computational complexity but at a cost of memory. Essentially what is needed is more capable hardware or smarter algorithms.

Here are some methods:

• Compression of the volume of sets of quantum states of the Qubit's metric:

The Fisher information metric can be used to map the volume of the qubit using an information geometry approach as discussed in "The Volume of Two-Qubit States by Information Geometry", "Analysis of Fisher Information and the Cramer-Rao Bound for Nonlinear Parameter Estimation After Compressed Sensing", and our "Intuitive explanation of Fisher Information and Cramer-Rao bound".

• Analogous to operand compression:

Computing depth-optimal decompositions of logical operations: "A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits" or this Quora discussion on "Encoding the dimension of the particle".

• Analogous to memory compression:

Qutrit factorization using ternary arithmetic: "Factoring with Qutrits: Shor's Algorithm on Ternary and Metaplectic Quantum Architectures" and "Quantum Ternary Circuit Synthesis Using Projection Operations".

• Analogous to traditional optimization

"A Quantum Algorithm for Finding Minimum Exclusive-Or Expressions".

• Other:

Krull Dimensions or axiomatisation and graph rewriting: "Completeness of the ZX-calculus for Pure Qubit Clifford+T Quantum Mechanics".

By combining those techniques you ought to be able to squeeze the foot into the shoe. That would permit emulation of larger systems on conventional processors, just don't ask me to explain doctoral level work or write the code. :)