With a combination of the quantum repeater and the cluster state approaches, we show that efficient quantum computation can be constructed even if all the entangling quantum gates only succeed with an arbitrarily small probability p. The required computational overhead scales efficiently both with 1/p and n, where n is the number of qubits in the computation. This approach provides an efficient way to combat noise in a class of quantum computation implementation schemes, where the dominant noise leads to probabilistic signaled errors with an error probability 1-p far beyond any threshold requirement.With a combination of the quantum repeater and the cluster state approaches, we show that efficient quantum computation can be constructed even if all the entangling quantum gates only succeed with an arbitrarily small probability p. The required computational overhead scales efficiently both with 1/p and n, where n is the number of qubits in the computation. This approach provides an efficient way to combat noise in a class of quantum computation implementation schemes, where the dominant noise leads to probabilistic signaled errors with an error probability 1-p far beyond any threshold requirement.

arXiv:quant-ph/0502120Physical Review LettersAug 17, 2005185 citations

Probabilistic Logics in Quantum Computation

Héctor Freytes, Antonio Ledda, Giuseppe Sergioli +1 more

Springer eBooksJan 1, 20134 citations

EP-PQM: Efficient Parametric Probabilistic Quantum Memory With Fewer Qubits and Gates

Mushahid Khan, Jean Paul Latyr Faye, Udson C. Mendes +1 more

Machine learning (ML) classification tasks can be carried out on a quantum computer (QC) using probabilistic quantum memory (PQM) and its extension, parametric PQM (P-PQM), by calculating the Hamming distance between an input pattern and a database of $r$ patterns containing $z$ features with $a$ distinct attributes. For PQM and P-PQM to correctly compute the Hamming distance, the feature must be encoded using one-hot encoding, which is memory intensive for multiattribute datasets with $a>2$ . We can represent multiattribute data more compactly by replacing one-hot encoding with label encoding; both encodings yield the same Hamming distance. Implementing this replacement on a classical computer is trivial. However, replacing these encoding schemes on a QC is not straightforward because PQM and P-PQM operate at the bit level, rather than at the feature level (a feature is represented by a binary string of 0’s and 1’s). We present an enhanced P-PQM, called efficient P-PQM (EP-PQM), that allows label encoding of data stored in a PQM data structure and reduces the circuit depth of the data storage and retrieval procedures. We show implementations for an ideal QC and a noisy intermediate-scale quantum (NISQ) device. Our complexity analysis shows that the EP-PQM approach requires $O(z \log _{2}(a))$ qubits as opposed to $O(za)$ qubits for P-PQM. EP-PQM also requires fewer gates, reducing gate count from $O(rza)$ to $O(rz\log _{2}(a))$ . For five datasets, we demonstrate that training an ML classification model using EP-PQM requires 48% to 77% fewer qubits than P-PQM for datasets with $a>2$ . EP-PQM reduces circuit depth in the range of 60% to 96%, depending on the dataset. The depth decreases further with a decomposed circuit, ranging between 94% and 99%. EP-PQM requires less space; thus, it can train on and classify larger datasets than previous PQM implementations on NISQ devices. Furthermore, reducing the number of gates speeds up the classification and reduces the noise associated with deep quantum circuits. Thus, EP-PQM brings us closer to scalable ML on an NISQ device.Machine learning (ML) classification tasks can be carried out on a quantum computer (QC) using probabilistic quantum memory (PQM) and its extension, parametric PQM (P-PQM), by calculating the Hamming distance between an input pattern and a database of$r$patterns containing$z$features with$a$distinct attributes. For PQM and P-PQM to correctly compute the Hamming distance, the feature must be encoded using one-hot encoding, which is memory intensive for multiattribute datasets with$a>2$. We can represent multiattribute data more compactly by replacing one-hot encoding with label encoding; both encodings yield the same Hamming distance. Implementing this replacement on a classical computer is trivial. However, replacing these encoding schemes on a QC is not straightforward because PQM and P-PQM operate at the bit level, rather than at the feature level (a feature is represented by a binary string of 0’s and 1’s). We present an enhanced P-PQM, called efficient P-PQM (EP-PQM), that allows label encoding of data stored in a PQM data structure and reduces the circuit depth of the data storage and retrieval procedures. We show implementations for an ideal QC and a noisy intermediate-scale quantum (NISQ) device. Our complexity analysis shows that the EP-PQM approach requires$O(z \log _{2}(a))$qubits as opposed to$O(za)$qubits for P-PQM. EP-PQM also requires fewer gates, reducing gate count from$O(rza)$to$O(rz\log _{2}(a))$. For five datasets, we demonstrate that training an ML classification model using EP-PQM requires 48% to 77% fewer qubits than P-PQM for datasets with$a>2$. EP-PQM reduces circuit depth in the range of 60% to 96%, depending on the dataset. The depth decreases further with a decomposed circuit, ranging between 94% and 99%. EP-PQM requires less space; thus, it can train on and classify larger datasets than previous PQM implementations on NISQ devices. Furthermore, reducing the number of gates speeds up the classification and reduces the noise associated with deep quantum circuits. Thus, EP-PQM brings us closer to scalable ML on an NISQ device.

arXiv:2201.07265IEEE Transactions on Quantum EngineeringJan 1, 20228 citations

Quantum computation: Algorithms and Applications

Chien‐Hung Cho, Chih-Yu Chen, Kuo-Chin Chen +6 more

Chinese Journal of PhysicsMay 8, 202148 citations

Probabilistic Interpolation of Quantum Rotation Angles

Bálint Koczor, John J. L. Morton, Simon C. Benjamin

Quantum computing requires a universal set of gate operations; regarding gates as rotations, any rotation angle must be possible. However a real device may only be capable of B bits of resolution, i.e., it might support only 2^{B} possible variants of a given physical gate. Naive discretization of an algorithm's gates to the nearest available options causes coherent errors, while decomposing an impermissible gate into several allowed operations increases circuit depth. Conversely, demanding higher B can greatly complexify hardware. Here, we explore an alternative: probabilistic angle interpolation (PAI). This effectively implements any desired, continuously parametrized rotation by randomly choosing one of three discretized gate settings and postprocessing individual circuit outputs. The approach is particularly relevant for near-term applications where one would in any case average over many runs of circuit executions to estimate expected values. While PAI increases that sampling cost, we prove that (a) the approach is optimal in the sense that PAI achieves the least possible overhead and (b) the overhead is remarkably modest even with thousands of parametrized gates and only seven bits of resolution available. This is a profound relaxation of engineering requirements for first generation quantum computers where even 5-6 bits of resolution may suffice and, as we demonstrate, the approach is many orders of magnitude more efficient than prior techniques. Moreover we conclude that, even for more mature late noisy intermediate-scale quantum era hardware, no more than nine bits will be necessary.Quantum computing requires a universal set of gate operations; regarding gates as rotations, any rotation angle must be possible. However a real device may only be capable of B bits of resolution, i.e., it might support only 2^{B} possible variants of a given physical gate. Naive discretization of an algorithm's gates to the nearest available options causes coherent errors, while decomposing an impermissible gate into several allowed operations increases circuit depth. Conversely, demanding higher B can greatly complexify hardware. Here, we explore an alternative: probabilistic angle interpolation (PAI). This effectively implements any desired, continuously parametrized rotation by randomly choosing one of three discretized gate settings and postprocessing individual circuit outputs. The approach is particularly relevant for near-term applications where one would in any case average over many runs of circuit executions to estimate expected values. While PAI increases that sampling cost, we prove that (a) the approach is optimal in the sense that PAI achieves the least possible overhead and (b) the overhead is remarkably modest even with thousands of parametrized gates and only seven bits of resolution available. This is a profound relaxation of engineering requirements for first generation quantum computers where even 5-6 bits of resolution may suffice and, as we demonstrate, the approach is many orders of magnitude more efficient than prior techniques. Moreover we conclude that, even for more mature late noisy intermediate-scale quantum era hardware, no more than nine bits will be necessary.

arXiv:2305.19881Physical Review LettersMar 27, 202417 citations

Simulations of Probabilities for Quantum Computing

Michail A. Zak

Jan 1, 19960 citations

r

patterns containing

z

features with

a

distinct attributes. For PQM and P-PQM to correctly compute the Hamming distance, the feature must be encoded using one-hot encoding, which is memory intensive for multiattribute datasets with

a>2

. We can represent multiattribute data more compactly by replacing one-hot encoding with label encoding; both encodings yield the same Hamming distance. Implementing this replacement on a classical computer is trivial. However, replacing these encoding schemes on a QC is not straightforward because PQM and P-PQM operate at the bit level, rather than at the feature level (a feature is represented by a binary string of 0’s and 1’s). We present an enhanced P-PQM, called efficient P-PQM (EP-PQM), that allows label encoding of data stored in a PQM data structure and reduces the circuit depth of the data storage and retrieval procedures. We show implementations for an ideal QC and a noisy intermediate-scale quantum (NISQ) device. Our complexity analysis shows that the EP-PQM approach requires

O(z \log _{2}(a))

qubits as opposed to

O(za)

qubits for P-PQM. EP-PQM also requires fewer gates, reducing gate count from

O(rza)

O(rz\log _{2}(a))

. For five datasets, we demonstrate that training an ML classification model using EP-PQM requires 48% to 77% fewer qubits than P-PQM for datasets with

a>2

. EP-PQM reduces circuit depth in the range of 60% to 96%, depending on the dataset. The depth decreases further with a decomposed circuit, ranging between 94% and 99%. EP-PQM requires less space; thus, it can train on and classify larger datasets than previous PQM implementations on NISQ devices. Furthermore, reducing the number of gates speeds up the classification and reduces the noise associated with deep quantum circuits. Thus, EP-PQM brings us closer to scalable ML on an NISQ device.