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The Design and Optimization of Quantum Circuits using the Palindrome Transform

Krysta M. Svore, Alfred V. Aho

Published Sep 1, 2003• 5 citations
OpenAlex

Abstract

The design and optimization of quantum circuits is central to quantum computation. This paper presents new algorithms for compiling arbitrary 2n×2n unitary matrices into efficient circuits of (n−1)-controlled single-qubit and (n−1)-controlled-NOT gates. We first present a general algebraic optimization technique, which we call the Palindrome Transform, that can be used to minimize the number of selfinverting gates in quantum circuits consisting of concatenations of palindromic subcircuits. For a fixed column ordering of two-level decomposition, we then give an enumerative algorithm for minimal (n−1)-controlled-NOT circuit construction, which we call the Palindromic Optimization Algorithm. Our work dramatically reduces the number of gates generated by the conventional two-level decomposition method for constructing quantum circuits of (n − 1)-controlled single-qubit and (n − 1)-controlled-NOT gates.

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Compiling Quantum Circuits using the Palindrome Transform

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The design and optimization of quantum circuits is central to quantum computation. This paper presents new algorithms for compiling arbitrary 2^n x 2^n unitary matrices into efficient circuits of (n-1)-controlled single-qubit and (n-1)-controlled-NOT gates. We first present a general algebraic optimization technique, which we call the Palindrome Transform, that can be used to minimize the number of self-inverting gates in quantum circuits consisting of concatenations of palindromic subcircuits. For a fixed column ordering of two-level decomposition, we then give an numerative algorithm for minimal (n-1)-controlled-NOT circuit construction, which we call the Palindromic Optimization Algorithm. Our work dramatically reduces the number of gates generated by the conventional two-level decomposition method for constructing quantum circuits of (n-1)-controlled single-qubit and (n-1)-controlled-NOT gates.

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We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum circuit representations of the algebraic generators of the relevant Lie algebras allowing the direct implementation of a Cartan decomposition on a quantum computer. The construction is recursive and allows us to expand any circuit down to generators and rotation matrices on individual qubits, where through our recursive algorithm we find that the generators themselves can be expressed with controlled-not (cnot) and swap gates explicitly. Our approach is independent of the standard cnot implementation and can be easily adapted to other cross-qubit circuit elements. In addition to its versatility, we also achieve near-optimal counts when working with cnot gates, achieving an asymptotic cnot cost of $\frac{21}{16}{4}^{n}$ for $n$ qubits.

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An arbitrary two-qubit computation in 23 elementary gates or less

Stephen S. Bullock, Igor L. Markov

Quantum circuits currently constitute a dominant model for quantum computation according to M. Nielsen and I. Chuang (2000). Our work addresses the problem of constructing quantum circuits to implement an arbitrary given quantum computation, in the special case of two qubits. We pursue circuits without ancilla qubits and a small number of elementary quantum gates by A. Barenco et al. (1995) and G. Song and A. Klappenecker (2003) as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2 CNOTs. To this end, we constructively prove a worst-case upper bound to 23 elementary gates, of which at most 4 (CNOTs) entail multi-qubit interactions. Our analysis shows that previous known synthesis algorithms, although more general, entail much larger quantum circuits than ours in the special case of two qubits. One such algorithm according to G. Cybenko (2001) has a worst case of 61 gates of which 18 may be CNOTs. Our technique rely on the KAK decomposition from Lie theory as well as the polar and spectral (symmetric Shur) matrix decompositions from numerical analysis. They are related to the canonical decomposition of two-qubit gate with respect to the "magic basis" of phase-shifted Bell states by N. Khaneja et al. (2001) and M. Lewenstein et al. (2001). We extend this decomposition in terms of elementary gates for quantum computation.

ACM/IEEE Design Automation Conference (DAC)•Mar 1, 2004•5 citations
The design and optimization of quantum circuits is central to quantum computation. This paper presents new algorithms for compiling arbitrary 2n×2n unitary matrices into efficient circuits of (n−1)-controlled single-qubit and (n−1)-controlled-NOT gates. We first present a general algebraic optimization technique, which we call the Palindrome Transform, that can be used to minimize the number of selfinverting gates in quantum circuits consisting of concatenations of palindromic subcircuits. For a fixed column ordering of two-level decomposition, we then give an enumerative algorithm for minimal (n−1)-controlled-NOT circuit construction, which we call the Palindromic Optimization Algorithm. Our work dramatically reduces the number of gates generated by the conventional two-level decomposition method for constructing quantum circuits of (n − 1)-controlled single-qubit and (n − 1)-controlled-NOT gates.
The design and optimization of quantum circuits is central to quantum computation. This paper presents new algorithms for compiling arbitrary 2^n x 2^n unitary matrices into efficient circuits of (n-1)-controlled single-qubit and (n-1)-controlled-NOT gates. We first present a general algebraic optimization technique, which we call the Palindrome Transform, that can be used to minimize the number of self-inverting gates in quantum circuits consisting of concatenations of palindromic subcircuits. For a fixed column ordering of two-level decomposition, we then give an numerative algorithm for minimal (n-1)-controlled-NOT circuit construction, which we call the Palindromic Optimization Algorithm. Our work dramatically reduces the number of gates generated by the conventional two-level decomposition method for constructing quantum circuits of (n-1)-controlled single-qubit and (n-1)-controlled-NOT gates.
Unitary decomposition is a widely used method to map quantum algorithms to an arbitrary set of quantum gates. Efficient implementation of this decomposition allows for the translation of bigger unitary gates into elementary quantum operations, which is key to executing these algorithms on existing quantum computers. The decomposition can be used as an aggressive optimization method for the whole circuit, as well as to test part of an algorithm on a quantum accelerator. For the selection and implementation of the decomposition algorithm, perfect qubits are assumed. We base our decomposition technique on Quantum Shannon Decomposition, which generates O(344n) controlled-not gates for an n-qubit input gate. In addition, we implement optimizations to take advantage of the potential underlying structure in the input or intermediate matrices, as well as to minimize the execution time of the decomposition. Comparing our implementation to Qubiter and the UniversalQCompiler (UQC), we show that our implementation generates circuits that are much shorter than those of Qubiter and not much longer than the UQC. At the same time, it is also up to 10 times as fast as Qubiter and about 500 times as fast as the UQC.
We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum circuit representations of the algebraic generators of the relevant Lie algebras allowing the direct implementation of a Cartan decomposition on a quantum computer. The construction is recursive and allows us to expand any circuit down to generators and rotation matrices on individual qubits, where through our recursive algorithm we find that the generators themselves can be expressed with controlled-not (cnot) and swap gates explicitly. Our approach is independent of the standard cnot implementation and can be easily adapted to other cross-qubit circuit elements. In addition to its versatility, we also achieve near-optimal counts when working with cnot gates, achieving an asymptotic cnot cost of 21164n\frac{21}{16}{4}^{n}1621​4n for nnn qubits.
In quantum computation every unitary operation can be decomposed into quantum circuits---a series of single-qubit rotations and a single type entangling two-qubit gates, such as controlled-not(cnot) gates. Two measures are important when judging the complexity of the circuit: the total number of cnot gates needed to implement it and the depth of the circuit, measured by the minimal number of computation steps needed to perform it. Here we give an explicit and simple quantum circuit scheme for preparation of arbitrary quantum states, which can directly utilize any decomposition scheme for arbitrary full quantum gates, thus connecting the two problems. Our circuit reduces the depth of the best currently known circuit by a factor of 222. It also reduces the total number of cnot gates from 2n{2}^{n}2n to 23242n\frac{23}{24}{2}^{n}2423​2n in the leading order for even number of qubits. Specifically, the scheme allows us to decrease the upper bound from 111111 cnot gates to 999 and the depth from 111111 to 555 steps for four qubits. Our results are expected to help in designing and building small-scale quantum circuits using present technologies.
Currently, computing paradigm is ruled by the laws of classical physics and will continue till the dimensions of transistors reaches the size of atomic particles. Quantum computing (QC) is a new technology that employs laws of quantum mechanics to address issues such as irreversibility and power dissipation that are outside the scope of conventional computing models. QC offers a powerful stage for solving complex and compound problems like searching and number factoring. A critical task in utilizing quantum physics in many application fields is circuit design using reversible quantum gates. Unitary decomposition is a popular scheme for mapping quantum algorithms to any set of elementary quantum operations. Using decomposition techniques enables transformation of unitary matrices into fundamental quantum gates, which is critical for running algorithms on prevailing quantum computers. Any 3x3 reversible quantum gate can be decomposed into single-qubit rotation gates and two qubit CNOT gates. This paper is the first to present quantum implementations of FRSG1 and JTF1 gates into CNOT gates and single qubit U3 gates with different optimization levels on a platform provided by IBM. These gates are important in a variety of practical applications like Stochastic computing and parity generation circuits. In FRSG1 the implementation count of single qubit gates decreases by 56% and the count of two qubit gates by 15% after optimization, whereas in JTF1 gate the single qubit gate count reduces by 71% and the number of two qubit gates reduces by 41%.
Quantum circuits currently constitute a dominant model for quantum computation according to M. Nielsen and I. Chuang (2000). Our work addresses the problem of constructing quantum circuits to implement an arbitrary given quantum computation, in the special case of two qubits. We pursue circuits without ancilla qubits and a small number of elementary quantum gates by A. Barenco et al. (1995) and G. Song and A. Klappenecker (2003) as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2 CNOTs. To this end, we constructively prove a worst-case upper bound to 23 elementary gates, of which at most 4 (CNOTs) entail multi-qubit interactions. Our analysis shows that previous known synthesis algorithms, although more general, entail much larger quantum circuits than ours in the special case of two qubits. One such algorithm according to G. Cybenko (2001) has a worst case of 61 gates of which 18 may be CNOTs. Our technique rely on the KAK decomposition from Lie theory as well as the polar and spectral (symmetric Shur) matrix decompositions from numerical analysis. They are related to the canonical decomposition of two-qubit gate with respect to the "magic basis" of phase-shifted Bell states by N. Khaneja et al. (2001) and M. Lewenstein et al. (2001). We extend this decomposition in terms of elementary gates for quantum computation.