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arXiv:0907.1764 · Physical Review A

Efficient compression of quantum information

Martin Plesch, Vladimír Bužek

Published Mar 18, 2010• 49 citations
arXivHTMLPDF

Abstract

We propose a scheme for an exact efficient transformation of a tensor product state of many identically prepared qubits into a state of a logarithmically small number of qubits. Using a quadratic number of elementary quantum gates we transform $N$ identically prepared qubits into a state, which is nontrivial only on the first $\ensuremath{\lceil}\mathrm{log}{}_{2}(N+1)\ensuremath{\rceil}$ qubits. This procedure might be useful for quantum memories, as only a small portion of the original qubits has to be stored. Another possible application is in communicating a direction encoded in a set of quantum states, as the compressed state provides a high-effective method for such an encoding.

DOI
10.1103/physreva.81.032317

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We propose a scheme for an exact efficient transformation of a tensor product state of many identically prepared qubits into a state of a logarithmically small number of qubits. Using a quadratic number of elementary quantum gates we transform NNN identically prepared qubits into a state, which is nontrivial only on the first \ensuremath⌈log2(N+1)\ensuremath⌉\ensuremath{\lceil}\mathrm{log}{}_{2}(N+1)\ensuremath{\rceil}\ensuremath⌈log2​(N+1)\ensuremath⌉ qubits. This procedure might be useful for quantum memories, as only a small portion of the original qubits has to be stored. Another possible application is in communicating a direction encoded in a set of quantum states, as the compressed state provides a high-effective method for such an encoding.
We present one-shot compression protocols that optimally encode ensembles of N identically prepared mixed states into O(logN) qubits. In contrast to the case of pure-state ensembles, we find that the number of encoding qubits drops down discontinuously as soon as a nonzero error is tolerated and the spectrum of the states is known with sufficient precision. For qubit ensembles, this feature leads to a 25% saving of memory space. Our compression protocols can be implemented efficiently on a quantum computer.
Quantum data compression is of great significance to both quantum computation and quantum communication due to the limited resources for quantum storage. Here in this study, we present an efficient multi-qubit quantum data compression protocol in which N identically prepared qubits can be perfectly compressed into qubits. Moreover, the feasible implementation of the quantum data compression is also designed based on the IBM Quantum Experience. The experiment reveals efficient information compression performance using the current scheme. We believe this method might be useful in quantum information which one could make the encoding of information on the qubits more efficiently, and also reducing the resources consumption for quantum storage.
Identical preparation creates permutation symmetry that can be used for lossless quantum compression. For nnn copies of an unknown ddd-dimensional pure state, the tensor-power input lies in the fully symmetric subspace, whose dimension is polynomial in nnn for fixed local dimension. Schur--Weyl duality isolates this subspace coherently, allowing the fixed representation labels to be discarded while retaining all information needed for recovery. The resulting encoder and decoder are exact on the input family. Because tensor-power states span the symmetric subspace, the achieved memory dimension is also necessary for reversible coherent compression, so the scheme is space-optimal. The encoder is implemented recursively with Clebsch--Gordan transforms, and only the branches reached by symmetric inputs need to be reproduced. Standard efficient synthesis of these transforms yields polynomial-size circuits for fixed local dimension and arbitrary target accuracy. Thus identically prepared pure states in arbitrary dimension admit lossless, space-optimal compression with an efficient circuit implementation.
We show that the quantum Fisher information about any phase parameter encoded in a family of pure quantum states can be faithfully compressed into a single qubit, accompanied by a logarithmic amount of classical bits. When the phase is encoded into many identical copies of a qubit state on the equator of the Bloch sphere, we show that the compression can be implemented sequentially, by iteratively compressing pairs of qubits into a single qubit. We experimentally demonstrate this building block in a photonic setup, developing two alternative compression strategies, based on Type-I fusion gate and a postselected implementation of the CNOT gate.
Quantum state preparation is a central primitive in many quantum algorithms, yet it is generally resource intensive, with efficient constructions known only for structured families of states. This work introduces a method for preparing quantum states whose amplitudes are given by a degree−d-d−d polynomial, using circuits with logarithmic depth in the number nnn of qubits and only O(n)\mathcal O(n)O(n) ancilla qubits, improving previous approaches that required linear-depth circuits. The construction first relies on a block-encoding of an affine diagonal operator based on its Pauli-basis decomposition, which involves only nnn terms. A modified linear-combination-of-unitaries (LCU) technique is introduced to implement this decomposition in logarithmic depth, together with a novel circuit for the EXACT-one oracle that flags basis states in which exactly one qubit is in the state ∣1⟩|1\rangle∣1⟩. It then uses a generalized quantum eigenvalue transformation (GQET) to promote this affine operator to an arbitrary degree polynomial. Theoretical analysis and numerical simulations are reported along with a proof-of-principle implementation on a trapped-ion quantum processor using 141414 qubits and more than 500500500 primitive quantum gates. Because polynomial approximations are ubiquitous in scientific computing, this construction provides a scalable and resource-efficient approach to quantum state preparation, further improving the potential of quantum algorithms in fields such as chemistry, physics, engineering, and finance.
Abstract Quantum algorithms and simulations often require the preparation of complex states through sequences of 2-qubit gates. For a generic quantum state, the number of required gates grows exponentially with the number of qubits, becoming unfeasible on near-term quantum devices. Here, we aim at creating an approximate encoding of the target state using a limited number of gates. As a first step, we consider a quantum state that is efficiently represented classically, such as a one-dimensional matrix product state. Using tensor network techniques, we develop and implement an efficient optimization algorithm that approaches the optimal implementation, requiring a polynomial number of iterations. We, next, consider the implementation of the proposed optimization algorithm directly on a quantum computer and overcome inherent barren plateaus by employing a local cost function. Our work offers a universal method to prepare target states using local gates and represents a significant improvement over known strategies.