Graph states, which include for example Bell states, GHZ states and cluster states, form a well-known class of quantum states with applications ranging from quantum networks to error-correction. Deciding whether two graph states are equivalent up to single-qubit Clifford operations is known to be decidable in polynomial time and have been studied both in the context of producing certain required states in a quantum network but also in relation to stabilizer codes. The reason for the latter this is that single-qubit Clifford equivalent graph states exactly corresponds to equivalent stabilizer codes. We here consider the computational complexity of, given a graph state |G>, counting the number of graph states, single-qubit Clifford equivalent to |G>. We show that this problem is #P-Complete. To prove our main result we make use of the notion of isotropic systems in graph theory. We review the definition of isotropic systems and point out their strong relation to graph states. We believe that these isotropic systems can be useful beyond the results presented in this paper.
We show that qubit stabilizer states can be represented by non-negative quasi-probability distributions associated with a Wigner-Weyl-Moyal formalism where Clifford gates are positive state-independent maps. This is accomplished by generalizing the Wigner-Weyl-Moyal formalism to three generators instead of two---producing an exterior, or Grassmann, algebra---which results in Clifford group gates for qubits that act as a permutation on the finite Weyl phase space points naturally associated with stabilizer states. As a result, a non-negative probability distribution can be associated with each …
The ZX-calculus is a graphical calculus for reasoning about pure state qubit quantum mechanics. It is complete for pure qubit stabilizer quantum mechanics, meaning any equality involving only stabilizer operations that can be derived using matrices can also be derived pictorially. Stabilizer operations include the unitary Clifford group, as well as preparation of qubits in the state |0>, and measurements in the computational basis. For general pure state qubit quantum mechanics, the ZX-calculus is incomplete: there exist equalities involving non-stabilizer unitary operations on single qubits w…
We define a normal form for Clifford circuits, and we prove that every Clifford operator has a unique normal form. Moreover, we present a rewrite system by which any Clifford circuit can be reduced to normal form. This yields a presentation of Clifford operators in terms of generators and relations.
We present an algorithm, along with its implementation that finds T-optimal approximations of single-qubit Z-rotations using quantum circuits consisting of Clifford and T gates. Our algorithm is capable of handling errors in approximation down to size $10^{-15}$, resulting in optimal single-qubit circuit designs required for implementation of scalable quantum algorithms. Our implementation along with the experimental results are available in the public domain.
Clifford group lies at the core of quantum computation -- it underlies quantum error correction, its elements can be used to perform magic state distillation and they form randomized benchmarking protocols, Clifford group is used to study quantum entanglement, and more. The ability to utilize Clifford group elements in practice relies heavily on the efficiency of their circuit-level implementation. Finding short circuits is a hard problem; despite Clifford group being finite, its size grows quickly with the number of qubits $n$, limiting known optimal implementations to $n{=}4$ qubits. For $n{…
We examine the following problem: given a collection of Clifford gates, describe the set of unitaries generated by circuits composed of those gates. Specifically, we allow the standard circuit operations of composition and tensor product, as well as ancillary workspace qubits as long as they start and end in states uncorrelated with the input, which rule out common "magic state injection" techniques that make Clifford circuits universal. We show that there are exactly 57 classes of Clifford unitaries and present a full classification characterizing the gate sets which generate them. This is th…
In a continuous-variable optical system, the Gottesman-Kitaev-Preskill (GKP) qubit is a promising candidate for fault-tolerant quantum computation. To implement non-Clifford operations on GKP qubits, non-Gaussian operations are required. In this context, the implementation of a cubic phase gate by combining nonlinear feedforward with ancillary states has been widely researched. Recently, however, it is pointed out that the cubic phase gate is not the most suitable for non-Clifford operations on GKP qubits. In this work, we show that we can achieve linear optical implementation of non-Clifford …
Quantum Accreditation with Non-Clifford Two-qubit Gates
School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；Department of Physics, University of Warwick（华威大学物理系）；National Quantum Computing Centre, Rutherford Appleton Laboratory（拉瑟福德阿普尔顿实验室国家量子计算中心）
We consider four-dimensional qudits as qubit pairs and their qudit Pauli operators as qubit Clifford operators. This introduces a nesting, $C_1^2 \subset C_2^4 \subset C_3^2$, where $C_n^m$ is the $n$th level of the $m$-dimensional qudit Clifford hierarchy. If we can convert between logical qubits and qudits, then qudit Clifford operators are qubit non-Clifford operators. Conversion is achieved by qubit fusion and qudit fission using stabilizer circuits that consume a resource state. This resource is a fused qubit stabilizer state with a fault-tolerant state preparation using stabilizer circui…
We explore the implementation of pseudo-random single-qubit rotations and multi-qubit pseudo-random circuits constructed only from Clifford gates and the T-gate, a phase rotation of pi/4. Such a gate set would be appropriate for computations performed in a fault tolerant setting. For single-qubit rotations the distribution of parameters found for unitaries constructed from Clifford plus T quickly approaches that of random rotations and require significantly fewer gates than the construction of arbitrary single-qubit rotations. For Clifford plus T pseudo-random circuits we find an exponential c…
Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano's results, simplifying some proofs along the way. We also point out some corollaries to Matsumoto and Amano's work, including an intrinsic characterization of the Clifford+T subgroup of SO(3), which also yields an efficient T-optimal exact single-qubit synthesis algorithm. Interestingly, this also gives an alternative proof…
We describe a new method for the decomposition of an arbitrary $n$ qubit operator with entries in $\mathbb{Z}[i,\frac{1}{\sqrt{2}}]$, i.e., of the form $(a+b\sqrt{2}+i(c+d\sqrt{2}))/{\sqrt{2}^{k}}$, into Clifford+$T$ operators where $n\le 2$. This method achieves a bound of $O(k)$ gates using at most one ancilla using decomposition into $1$- and $2$-level matrices which was first proposed by Giles and Selinger.
Equivariant Reinforcement Learning for Clifford Quantum Circuit Synthesis
Department of Computer Science, University of Oxford（牛津大学计算机科学系）；Quantinuum（Quantinuum公司）；College of Computing and Data Science, Nanyang Technological University, Singapore（新加坡南洋理工大学计算科学与数据科学学院）
Benchmarking methods that can be adapted to multi-qubit systems are essential for assessing the overall or "holistic" performance of nascent quantum processors. The current industry standard is Clifford randomized benchmarking (RB), which measures a single error rate that quantifies overall performance. But scaling Clifford RB to many qubits is surprisingly hard. It has only been performed on 1, 2, and 3 qubits as of this writing. This reflects a fundamental inefficiency in Clifford RB: the $n$-qubit Clifford gates at its core have to be compiled into large circuits over the 1- and 2-qubit gat…
We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov and Mosca. Their algorithm takes as input an exactly synthesizable single-qubit unitary--one which can be expressed without error as a product of Clifford and T gates--and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer $n$ we consider the "Clifford-cyclotomic" gate set co…
The Clifford group is the set of gates generated by the controlled not gates, the Hadamard gate and the P={{1,0},{0,i}} gate. We will say that a n-qubit state is a Clifford state if it can be prepared using Clifford gates. In this paper we study the set of all 4-qubit Clifford states. We prove that there are 293760 states and their entanglement entropy must be either 0, 2/3, 1, 4/3 and 5/3. We also show that any pair of these states can be connected using local gates and at most 3 CNOT gates. We achieve this by splitting the 293760 states into 18 groups where each pair of states in a group can…
In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$, in the single-qubit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Hadamard and T gates used. We conjecture that the equivalence of the sets of unitaries implementable by circuits over the Clifford and T library and unitaries over the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$ holds in the $n$-qubit case.
The Clifford group is a fundamental structure in quantum information with a wide variety of applications. We discuss the tensor representations of the $q$-qubit Clifford group, which is defined as the normalizer of the $q$-qubit Pauli group in $U(2^q)$. In particular, we characterize all irreducible subrepresentations of the two-copy representation $φ^{\otimes2}$ of the Clifford group on the matrix space $\mathbb{C}^{d\times d}\otimes \mathbb{C}^{d\times d}$ with $d=2^q$. In an upcoming companion paper we applied this result to cut down the number of samples necessary to perform randomised ben…
ZX-calculus is a high-level graphical formalism for qubit computation. In this paper we give the ZX-rules that enable one to derive all equations between 2-qubit Clifford+T quantum circuits. Our rule set is only a small extension of the rules of stabilizer ZX-calculus, and substantially less than those needed for the recently achieved universal completeness. One of our rules is new, and we expect it to also have other utilities. These ZX-rules are much simpler than the complete of set Clifford+T circuit equations due to Selinger and Bian, which indicates that ZX-calculus provides a more conven…
Representations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of superfluous branches. The usual Jordan--Wigner construction also may be formally obtained in this approach by bringing the process up to trivial qubit chain (trunk). The methods can also be used for effective simulation of some quantum circuits corresponding to the binary tree structure. The modeling of more general qubit trees, as well as the relationship w…
Recently, we gave a complete axiomatisation of the ZX-calculus for the overall pure qubit quantum mechanics. Based on this result, here we also obtain a complete axiomatisation of the ZX-calculus for the Clifford+T quantum mechanics by restricting the ring of complex numbers to its subring corresponding to the Clifford+T fragment resting on the completeness theorem of the ZW-calculus for arbitrary commutative ring. In contrast to the first complete axiomatisation of the ZX-calculus for the Clifford+T fragment, we have two new generators as features rather than novelties: the triangle can be em…
We developed a general framework for synthesizing target gates by using a finite set of basic gates, which is a crucial step in quantum compilation. When approximating a gate in SU($n$), a naive brute-force search requires a computational complexity of $O(1/\varepsilon^{(n^2 - 1)})$ to achieve an approximation with error $\varepsilon$. In contrast, by using our method, the complexity can be reduced to $O(-n^2 \log\varepsilon/\varepsilon^{((n^2 - 1)/2)})$. This method requires almost no assumptions and can be applied to a variety of gate sets, including Clifford+$T$ and Clifford+$V$. Further, w…
Fault-tolerant quantum computation (FTQC) schemes using large block codes that encode $k>1$ qubits in $n$ physical qubits can potentially reduce the resource overhead to a great extent because of their high encoding rate. However, the fault-tolerant (FT) logical operations for the encoded qubits are difficult to find and implement, which usually takes not only a very large resource overhead but also long $\textit{in-situ}$ computation time. In this paper, we focus on Calderbank-Shor-Steane $[\![ n,k,d ]\!]$ (CSS) codes and their logical FT Clifford circuits. We show that the depth of an arbitr…
The Clifford group is a finite subgroup of the unitary group generated by the Hadamard, the CNOT, and the Phase gates. This group plays a prominent role in quantum error correction, randomized benchmarking protocols, and the study of entanglement. Here we consider the problem of finding a short quantum circuit implementing a given Clifford group element. Our methods aim to minimize the entangling gate count assuming all-to-all qubit connectivity. First, we consider circuit optimization based on template matching and design Clifford-specific templates that leverage the ability to factor out Pau…
We present an algorithm for building a circuit that approximates single qubit unitaries with precision ε using O(log(1/ε)) Clifford and T gates and employing up to two ancillary qubits. The algorithm for computing our approximating circuit requires an average of O(log^2(1/ε)log log(1/ε)) operations. We prove that the number of gates in our circuit saturates the lower bound on the number of gates required in the scenario when a constant number of ancillae are supplied, and as such, our circuits are asymptotically optimal. This results in significant improvement over the current state of the art…
Quantum gates in experiment are inherently prone to errors that need to be characterized before they can be corrected. Full characterization via quantum process tomography is impractical and often unnecessary. For most practical purposes, it is enough to estimate more general quantities such as the average fidelity. Here we use a unitary 2-design and twirling protocol for efficiently estimating the average fidelity of Clifford gates, to certify a 7-qubit entangling gate in a nuclear magnetic resonance quantum processor. Compared with more than $10^8$ experiments required by full process tomogr…
We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS=diag(1,1,1,i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford gates are typically more expensive to perform in a fault-tolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for two-qubit Clifford+C…
We give a general proof for the existence and realizability of Clifford gates in the Ising topological quantum computer. We show that all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates. We find that the braiding gates for two qubits exhaust the entire two-qubit Clifford group. Analyzing the structure of the Clifford group for n \geq 3 qubits we prove that the the image of the braid group is a non-trivial subgroup of the Clifford group so that not all Clifford gates could be implemented by braiding in the Ising topological quantum computation scheme. We als…
Clifford circuits -- i.e. circuits composed of only CNOT, Hadamard, and $π/4$ phase gates -- play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) -- where one additionally conjugates every qubit by the same one-qubit gate $U$ -- can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford c…
We present a planar surface-code-based scheme for fault-tolerant quantum computation which eliminates the time overhead of single-qubit Clifford gates, and implements long-range multi-target CNOT gates with a time overhead that scales only logarithmically with the control-target separation. This is done by replacing hardware operations for single-qubit Clifford gates with a classical tracking protocol. Inter-qubit communication is added via a modified lattice surgery protocol that employs twist defects of the surface code. The long-range multi-target CNOT gates facilitate magic state distillat…
University of Amsterdam（阿姆斯特丹大学）；QuSoft；Charles University, Prague, Czech Republic（布拉格查理大学）；Department of Computer Science, University of Oxford（牛津大学计算机系）
Hardware efficient transpilation of quantum circuits to a quantum devices native gateset is essential for the execution of quantum algorithms on noisy quantum computers. Typical quantum devices utilize a gateset with a single two-qubit Clifford entangling gate per pair of coupled qubits, however, in some applications access to a non-Clifford two-qubit gate can result in more optimal circuit decompositions and also allows more flexibility in optimizing over noise. We demonstrate calibration of a low error non-Clifford Controlled-$\fracπ{2}$ phase (CS) gate on a cloud based IBM Quantum computing…
We give an efficient randomized algorithm for approximating an arbitrary element of $SU(2)$ by a product of Clifford+$T$ operators, up to any given error threshold $ε>0$. Under a mild hypothesis on the distribution of primes, the algorithm's expected runtime is polynomial in $\log(1/ε)$. If the operator to be approximated is a $z$-rotation, the resulting gate sequence has $T$-count $K+4\log_2(1/ε)$, where $K$ is approximately equal to $10$. We also prove a worst-case lower bound of $K+4\log_2(1/ε)$, where $K=-9$, so that our algorithm is within an additive constant of optimal for certain $z$-r…
We provide a purely quantum version of polar codes, achieving the symmetric coherent information of any qubit-input quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where a two-qubit gate randomly chosen from the Clifford group is used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by preshared EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that quantum polarization can be achieved by choosing the channel combining Clifford operator randomly, from …
