# Papers

## Preprints

### The equivalence between correctability of deletions and insertions of separable states in quantum codes

### Taro Shibayama, Yingkai Ouyang

In this paper, we prove the equivalence of inserting separable quantum states and deletions. Hence any quantum code that corrects deletions automatically corrects separable insertions. First, we describe the quantum insertion/deletion error using the Kraus operators. Next, we develop an algebra for commuting Kraus operators corresponding to insertions and deletions. Using this algebra, we prove the equivalence between quantum insertion codes and quantum deletion codes using the Knill-Laflamme conditions.

### Yingkai Ouyang

Quantum deletions, which are harder to correct than erasure errors, occur in many realistic settings. It is therefore pertinent to develop quantum coding schemes for quantum deletion channels. To date, not much is known about which explicit quantum error correction codes can combat quantum deletions. We note that *any* permutation-invariant quantum code that has a distance of t+1 can correct t quantum deletions for any positive integer t in both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corresponding coding bounds for permutation-invariant quantum codes under quantum deletions. We focus our attention on a specific family of N-qubit permutation-invariant quantum codes, which we call shifted gnu codes, and show that their encoding and decoding algorithms can be performed in O(N) and O(N^2).

### Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians

### Ramis Movassagh, Yingkai Ouyang

We introduce a framework for constructing a quantum error correcting code from any classical error correcting code. This includes CSS codes and goes beyond the stabilizer formalism to allow quantum codes to be constructed from classical codes that are not necessarily linear or self-orthogonal (Fig. 1). We give an algorithm that explicitly constructs quantum codes with linear distance and constant rate from classical codes with a linear distance and rate. As illustrations for small size codes, we obtain Steane's 7−qubit code uniquely from Hamming's [7,4,3] code, and obtain other error detecting quantum codes from other explicit classical codes of length 4 and 6. Motivated by quantum LDPC codes and the use of physics to protect quantum information, we introduce a new 2-local frustration free quantum spin chain Hamiltonian whose ground space we analytically characterize completely. By mapping classical codewords to basis states of the ground space, we utilize our framework to demonstrate that the ground space contains explicit quantum codes with linear distance. This side-steps the Bravyi-Terhal no-go theorem because our work allows for more general quantum codes beyond the stabilizer and/or linear codes. We hesitate to call this an example of *subspace* quantum LDPC code with linear distance.

### Yingkai Ouyang and Earl Campbell

Minimizing the number of particles used by a quantum code is helpful, because every particle incurs a cost. One quantum error correction solution is to encode quantum information into one or more bosonic modes. We revisit rotation-invariant bosonic codes, which are supported on Fock states that are gapped by an integer *g* apart, and the gap imparts number shift resilience to these codes. Intuitively, since phase operators and number shift operators do not commute, one expects a trade-off between resilience to number-shift and rotation errors. Here, we obtain results pertaining to the non-existence of approximate quantum error correcting *g*-gapped single-mode bosonic codes with respect to Gaussian dephasing errors. We show that by using arbitrarily many modes, *g*-gapped multi-mode codes can yield good approximate quantum error correction codes for any finite magnitude of Gaussian dephasing errors.

### Yingkai Ouyang, Narayanan Rengaswamy

Quantum metrology (QM) is expected to be a prominent use-case of quantum technologies. However, noise easily degrades these quantum probe states, and negates the quantum advantage they would have offered in a noiseless setting. Although quantum error correction (QEC) can help tackle noise, fault-tolerant methods are too resource intensive for near-term use. Hence, a strategy for (near-term) robust QM that is easily adaptable to future QEC-based QM is desirable. Here, we propose such an architecture by studying the performance of quantum probe states that are constructed from [n,k,d] binary block codes of minimum distance d≥t+1. Such states can be interpreted as a logical state of a CSS code whose logical X group is defined by the aforesaid binary code. When a constant, t, number of qubits of the quantum probe state are erased, using the quantum Fisher information (QFI) we show that the resultant noisy probe can give an estimate of the magnetic field with a precision that scales inversely with the variances of the weight distributions of the corresponding 2t shortened codes. If C is any code concatenated with inner repetition codes of length linear in n, a quantum advantage in QM is possible. Hence, given any CSS code of constant length, concatenation with repetition codes of length linear in n is asymptotically optimal for QM with a constant number of erasure errors. We also explicitly construct an observable that when measured on such noisy code-inspired probe states, yields a precision on the magnetic field strength that also exhibits a quantum advantage in the limit of vanishing magnetic field strength. We emphasize that, despite the use of coding-theoretic methods, our results do not involve syndrome measurements or error correction. We complement our results with examples of probe states constructed from Reed-Muller codes.

### Yingkai Ouyang, N. Shettell, D. Markham

Quantum metrology is a promising practical use case for quantum technologies, where physical quantities can be measured with unprecedented precision. In lieu of quantum error correction procedures, near term quantum devices are expected to be noisy, and we have to make do with noisy probe states. With carefully chosen symmetric probe states inspired by the quantum error correction capabilities of certain symmetric codes, we prove that quantum metrology can exhibit an advantage over classical metrology, even after the probe states are corrupted by a constant number of erasure and dephasing errors. These probe states prove useful for robust metrology not only in the NISQ regime, but also in the asymptotic setting where they achieve Heisenberg scaling. This brings us closer towards making robust quantum metrology a technological reality.

## Publications

### 23. Avoiding coherent errors with rotated concatenated stabilizer codes npj Quantum Information 7 (87)

### Yingkai Ouyang

Coherent errors, which arise from collective couplings, are a dominant form of noise in many realistic quantum systems, and are more damaging than oft considered stochastic errors. Here, we propose integrating stabilizer codes with coherent-error-avoiding codes by code concatenation. Namely, by concatenating an [[n,k,d]] stabilizer outer code with dual-rail inner codes, we obtain a [[2n,k,d]] non-stabilizer constant-excitation code immune from coherent phase errors and also equivalent to a Pauli-rotated stabilizer code. When the stabilizer outer code is fault-tolerant, the constant-excitation code has a positive fault-tolerant threshold against stochastic errors. Setting the outer code as a four-qubit amplitude damping code yields an eight-qubit constant-excitation code that corrects a single amplitude damping error, and we analyze this code's potential as a quantum memory.

### Yingkai Ouyang

Quantum data is inherently fragile and must be protected to unlock the potential of quantum technologies. A pertinent concern in schemes for quantum storage is their potential to be implemented in the near future. While Heisenberg ferromagnets are readily available and can be potentially implemented, quantum storage in them has never before been addressed. We address this issue by considering the storage of quantum data within a special quantum ferromagnet, where every pair of spins interacts with equal strength. We analyze the storage error for a unital and local noise model, and optimize the memory lifetime with respect to system size. Our analysis relies on Taylor decompositions of unitary evolutions in terms of Fréchet matrix derivatives, and uses Davis' divided difference representation for these Fréchet derivatives, and the recursive structure of these divided differences. We thereby obtain upper bounds on the error for passive quantum storage. With our bounds, we numerically study the potential to enhance memory lifetimes. Our approach lays the foundation for optimization of the memory lifetime based on the spectral structure of any physical system.

### 21. Quantifying quantum speedups: improved classical simulation from tighter magic monotones, PRX Quantum 2 (1), 010345

### James R. Seddon, Bartosz Regula, Hakop Pashayan, Yingkai Ouyang, Earl T. Campbell

In the stabilizer circuit model of quantum computation, universality requires a resource known as magic. Here, we propose three new ways to quantify the magic of a quantum state using magic monotones, apply the monotones in the characterization of state conversions under stabilizer operations, and connect them with the classical simulation of quantum circuits. We first present a complete theory of these quantifiers for tensor products of single-qubit states, for which the monotones are all equal and all act multiplicatively, constituting the first qubit magic monotones to have this property. We use the monotones to establish several asymptotic and non-asymptotic bounds on state interconversion and distillation rates. We then relate our quantifiers directly to the runtime of classical simulation algorithms, showing that a large amount of magic is a necessary requirement for any quantum speedup. One of our classical simulation algorithms is a quasi-probability simulator with its runtime connected to a generalized notion of negativity, which is exponentially faster than all prior qubit quasi-probability simulation algorithms. We also introduce a new variant of the stabilizer rank simulation algorithm suitable for mixed states, while improving the runtime bounds for this class of simulations. Our work reveals interesting connections between quasi-probability and stabilizer rank simulators, which previously appeared to be unrelated. Generalizing the approach beyond the theory of magic states, we establish methods for the quantitative characterization of classical simulability for more general quantum resources, and use them in the resource theory of quantum coherence to connect the ℓ1-norm of coherence with the simulation of free operations.

### 20. Tight bounds on the simultaneous estimation of incompatible parameters, Physical Review X 11, 011028

### Jasminder S. Sidhu, Yingkai Ouyang, Earl T. Campbell, Pieter Kok

The estimation of multiple parameters in quantum metrology is important for a vast array of applications in quantum information processing. However, the unattainability of fundamental precision bounds for incompatible observables has greatly diminished the applicability of estimation theory in many practical implementations. The Holevo Cramer-Rao bound (HCRB) provides the most fundamental, simultaneously attainable bound for multi-parameter estimation problems. A general closed form for the HCRB is not known given that it requires a complex optimisation over multiple variables. In this work, we show that the HCRB can be solved analytically for two parameters. For more parameters, we generate a lower bound to the HCRB. Our work greatly reduces the complexity of determining the HCRB to solving a set of linear equations. We apply our formalism to magnetic field sensing. Our results provide fundamental insight and make significant progress towards the estimation of multiple incompatible observables.

### 19. Linear programming bounds for quantum amplitude damping codes, 2020 IEEE International Symposium on Information Theory, arXiv:2001.03976

### Yingkai Ouyang, Ching-Yi Lai

Given that approximate quantum error-correcting (AQEC) codes have a potentially better performance than perfect quantum error correction codes, it is pertinent to quantify their performance. While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds do not directly apply to AQEC codes. Herein, we introduce quantum weight enumerators for amplitude damping (AD) errors and work within the framework of approximate quantum error correction. In particular, we introduce an auxiliary exact weight enumerator that is intrinsic to a code space and moreover, we establish a linear relationship between the quantum weight enumerators for AD errors and this auxiliary exact weight enumerator. This allows us to establish a linear program that is infeasible only when AQEC AD codes with corresponding parameters do not exist. To illustrate our linear program, we numerically rule out the existence of three-qubit AD codes that are capable of correcting an arbitrary AD error.

### 18. Homomorphic encryption of linear optics quantum computation on almost arbitrary states of light with asymptotically perfect security, Physical Review Research

### Yingkai Ouyang, Si-Hui Tan, Joesph F. Fitzsimons, Peter Rohde

Future quantum computers are likely to be expensive and affordable outright by few, motivating client/server models for outsourced computation. However, the applications for quantum computing will often involve sensitive data, and the client would like to keep her data secret, both from eavesdroppers and the server itself. Homomorphic encryption is an approach for encrypted, outsourced quantum computation, where the client's data remains secret, even during execution of the computation. We present a scheme for the homomorphic encryption of arbitrary quantum states of light with no more than a fixed number of photons, under the evolution of both passive and adaptive linear optics, the latter of which is universal for quantum computation. The scheme uses random coherent displacements in phase-space to obfuscate client data. In the limit of large coherent displacements, the protocol exhibits asymptotically perfect information-theoretic secrecy. The experimental requirements are modest, and easily implementable using present-day technology.

### Yingkai Ouyang, D.R White, E. Campbell

Simulation of quantum chemistry is expected to be a principal application of quantum computing. In quantum simulation, a complicated Hamiltonian describing the dynamics of a quantum system is decomposed into its constituent terms, where the effect of each term during time-evolution is individually computed. For many physical systems, the Hamiltonian has a large number of terms, constraining the scalability of established simulation methods. To address this limitation we introduce a new scheme that approximates the actual Hamiltonian with a sparser Hamiltonian containing fewer terms. By stochastically sparsifying weaker Hamiltonian terms, we benefit from a quadratic suppression of errors relative to deterministic approaches. Relying on optimality conditions from convex optimisation theory, we derive an appropriate probability distribution for the weaker Hamiltonian terms, and compare its error bounds with other probability ansatzes for some electronic structure Hamiltonians. Tuning the sparsity of our approximate Hamiltonians allows our scheme to interpolate between two recent random compilers: qDRIFT and randomized first order Trotter. Our scheme is thus an algorithm that combines the strengths of randomised Trotterisation with the efficiency of qDRIFT, and for intermediate gate budgets, outperforms both of these prior methods.

### 16. Faster quantum computation with permutations and resonant couplings, Linear Algebra and Applications

### Yingkai Ouyang, Y. Shen, L. Chen

Recently, there has been increasing interest in designing schemes for quantum computations that are robust against errors. Although considerable research has been devoted to developing quantum error correction schemes, much less attention has been paid to optimizing the speed it takes to perform a quantum computation and developing computation models that act on decoherence-free subspaces. Speeding up a quantum computation is important, because fewer errors are likely to result. Encoding quantum information in a decoherence-free subspace is also important, because errors would be inherently suppressed. In this paper, we consider quantum computation in a decoherence-free subspace and also optimize its speed. To achieve this, we perform certain single-qubit quantum computations by simply permuting the underlying qubits. Together with exchange-interactions or Ising-interactions and other resonant couplings, we present a new scheme for quantum computation that potentially improves the speed in which a quantum computation can be done.

### 15. Permutation-invariant constant-excitation quantum codes for amplitude damping, IEEE Transactions on Information Theory

### Yingkai Ouyang, Rui Chao

The increasing interest in using quantum error correcting codes in practical devices has heightened the need for designing quantum error correcting codes that can correct against specialized errors, such as that of amplitude damping errors which model photon loss. Although considerable research has been devoted to quantum error correcting codes for amplitude damping, not so much attention has been paid to having these codes simultaneously lie within the decoherence free subspace of their underlying physical system. One common physical system comprises of quantum harmonic oscillators, and constant-excitation quantum codes can be naturally stabilized within them. The purpose of this paper is to give constant-excitation quantum codes that not only correct amplitude damping errors, but are also immune against permutations of their underlying modes. To construct such quantum codes, we use the nullspace of a specially constructed matrix based on integer partitions.

### R. Pisarczyk, Z. Zhao, Yingkai Ouyang, J. F. Fitzsimons, V. Vedral

The capacity of a channel is known to be equivalent to the highest rate at which it can generate entanglement. Analogous to entanglement, the notion of quantum causality characterises the temporal aspect of quantum correlations. Despite holding an equally fundamental role in physics, temporal quantum correlations have yet to find their operational significance in quantum communication. Here we uncover a connection between quantum causality and channel capacity. We show the amount of temporal correlations between two ends of the noisy quantum channel, as quantified by a causality measure, implies a general upper bound on its channel capacity. The expression of this new bound is simpler to evaluate than most previously known bounds. We demonstrate the utility of this bound by applying it to a class of shifted depolarizing channels, which results in improvement over previously known bounds for this class of channels.

### 13. Computing spectral bounds of the Heisenberg ferromagnet from geometric considerations, Journal of Mathematical Physics

### Yingkai Ouyang

We give a polynomial-time algorithm for computing upper bounds on some of the smaller energy eigenvalues in a spin-1/2 ferromagnetic Heisenberg model with any graph G for the underlying interactions. An important ingredient is the connection between Heisenberg models and the symmetric products of G. Our algorithms for computing upper bounds are based on generalized diameters of graphs. Computing the upper bounds amounts to solving the minimum assignment problem on G, which has well-known polynomial-time algorithms from the field of combinatorial optimization. We also study the possibility of computing the lower bounds on some of the smaller energy eigenvalues of Heisenberg models. This amounts to estimating the isoperimetric inequalities of the symmetric product of graphs. By using connections with discrete Sobolev inequalities, we show that this can be performed by considering just the vertex-induced subgraphs of G. If our conjecture for a polynomial time approximation algorithm to solve the edge-isoperimetric problem holds, then our proposed method of estimating the energy eigenvalues via approximating the edge-isoperimetric properties of vertex-induced subgraphs will yield a polynomial time algorithm for estimating the smaller energy eigenvalues of the Heisenberg ferromagnet.

### C. Wu, Y. Wang, C. Guo, Yingkai Ouyang, G. Wang, XL Feng

Recently, there has been growing interest in using quantum error correction in practical devices. A central issue in quantum error correction is the initialization of quantum data into a quantum error-correction code. Most studies have concentrated on generating quantum codes based on their encoding quantum circuits. However, this often leads to a large number of steps required in the initialization, and hence this process can be prone to errors. The purpose of this work is to demonstrate that permutation-invariant quantum error-correction codes can be created with high fidelity by exploiting their underlying symmetry. The code is initialized on multiple qubits that mutually interact or are themselves coupled to a quantum harmonic oscillator. We show that the so-called selective resonant interaction is derivable on such physical systems. By utilizing the selective resonant interaction, these highly symmetric codes may be rapidly generated with excellent fidelity. We also discuss the potential of initializing permutation-invariant quantum error-correction codes based on the state-of-art experimental techniques.

### Yingkai Ouyang, S.H. Tan, J.F.Fitzsimons

The recent discovery of fully-homomorphic classical encryption schemes has had a dramatic effect on the direction of modern cryptography. Such schemes, however, implicitly rely on the assumptions that solving certain computation problems are intractable. Here we present a quantum encryption scheme which is homomorphic for arbitrary classical and quantum circuits which have at most some constant number of non-Clifford gates. Unlike classical schemes, the security of the scheme we present is information theoretic and hence independent of the computational power of an adversary.

### T.F. Demarie, Yingkai Ouyang, J.F. Fitzsimons

We consider the task of verifying the correctness of quantum computation for a restricted class of circuits which contain at most two basis changes. This contains circuits giving rise to the second level of the Fourier Hierarchy, the lowest level for which there is an established quantum advantage. We show that, when the circuit has an outcome with probability at least the inverse of some polynomial in the circuit size, the outcome can be checked in polynomial time with bounded error by a completely classical verifier. This verification procedure is based on random sampling of computational paths and is only possible given knowledge of the likely outcome.

### S.H. Tan, Yingkai Ouyang, P. Rohde

We present a scheme for implementing homomorphic encryption on coherent states encoded using phase-shift keys. The encryption operations require only rotations in phase space, which commute with computations in the codespace performed via passive linear optics, and with generalized non-linear phase operations that are polynomials of the photon-number operator in the codespace. This encoding scheme can thus be applied to any computation with coherent state inputs, and the computation proceeds via a combination of passive linear optics and generalized non-linear phase operations. An example of such a computation is matrix multiplication, whereby a vector representing coherent state amplitudes is multiplied by a matrix representing a linear optics network, yielding a new vector of coherent state amplitudes. By finding an orthogonal partitioning of the support of our encoded states, we quantify the security of our scheme via the indistinguishability of the encrypted codewords. Whilst we focus on coherent state encodings, we expect that this phase-key encoding technique could apply to any continuous-variable computation scheme where the phase-shift operator commutes with the computation.

### Yingkai Ouyang, S.H. Tan, L. Zhao, J. Fitzsimons

A (k,n)-threshold secret-sharing scheme allows for a string to be split into n shares in such a way that any subset of at least k shares suffices to recover the secret string, but such that any subset of at most k-1 shares contains no information about the secret. Quantum secret-sharing schemes extend this idea to the sharing of quantum states. Here we propose a method of performing computation on quantum shared secrets. We introduce a (n,n)-quantum secret sharing scheme together with a set of protocols that allow quantum circuits to be evaluated on the shared secret without the need to decode the secret. We consider a multipartite setting, with each participant holding a share of the secret. We show that if there exists at least one honest participant, no group of dishonest participants can recover any information about the shared secret, independent of their deviations from the protocol.

### Yingkai Ouyang

A permutation-invariant quantum code on N qudits is any subspace stabilized by the matrix representation of the symmetric group SN as permutation matrices that permute the underlying N subsystems. When each subsystem is a complex Euclidean space of dimension q >= 2, any permutation-invariant code is a subspace of the symmetric subspace N qudits of dimension q. We give an algebraic construction of new families of of d-dimensional permutation-invariant codes on at least (2t+1)2(d−1) qudits that can also correct t errors for d>=2. The construction of our codes relies on a real polynomial with multiple roots at the roots of unity, and a sequence of q−1 real polynomials that satisfy some combinatorial constraints. When N>(2t+1)2(d−1), we prove constructively that an uncountable number of such codes exist.

### S.H. Tan, J. Kettlewell, Yingkai Ouyang, L. Chen, J. Fitzsimons

Encryption schemes often derive their power from the properties of the underlying algebra on the symbols used. Inspired by group theoretic tools, we use the centralizer of a subgroup of operations to present a private-key quantum homomorphic encryption scheme that enables a broad class of quantum computation on encrypted data. A particular instance of our encoding hides up to a constant fraction of the information encrypted. This fraction can be made arbitrarily close to unity with overhead scaling only polynomially in the message length. This highlights the potential of our protocol to hide a non-trivial amount of information, and is suggestive of a large class of encodings that might yield better security.

### Yingkai Ouyang, J. F. Fitzsimons

permutation-invariant code on m qubits is a subspace of the symmetric subspace of the m qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading order spontaneous decay errors. To prove the result, we use elementary number theory with prior theory on permutation invariant codes and quantum error correction.

### Yingkai Ouyang

A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground-space contains all permutation-invariant states. We use techniques from combinatorics and operator theory to construct families of permutation-invariant quantum codes. These codes have length proportional to t2; one family of codes perfectly corrects arbitrary weight t errors, while the other family of codes approximately correct t spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.

### 3. Channel covariance, twirling, contraction, and some upper bounds on the quantum capacity, Quantum Information and Computation

### Yingkai Ouyang

Evaluating the quantum capacity of quantum channels is an important but difficult problem, even for channels of low input and output dimension. Smith and Smolin showed that the quantum capacity of the Clifford-twirl of a qubit amplitude damping channel (a qubit depolarizing channel) has a quantum capacity that is at most the coherent information of the qubit amplitude damping channel evaluated on the maximally mixed input state. We restrict our attention to obtaining upper bounds on the quantum capacity using a generalization of Smith and Smolin's degradable extension technique. Given a degradable channel N and a finite projective group of unitaries V, we show that the V-twirl of N has a quantum capacity at most the coherent information of N maximized over a V-contracted space of input states. As a consequence, degradable channels that are covariant with respect to diagonal Pauli matrices have quantum capacities that are their coherent information maximized over just the diagonal input states. As an application of our main result, we supply new upper bounds on the quantum capacity of some unital and non-unital channels -- d-dimensional depolarizing channels, two-qubit locally symmetric Pauli channels, and shifted qubit depolarizing channels.

### 2. Concatenated codes can attain the Quantum Gilbert-Varshamov bound, IEEE Transactions on Information Theory

### Yingkai Ouyang

A family of quantum codes of increasing block length with positive rate is asymptotically good if the ratio of its distance to its block length approaches a positive constant. The asymptotic quantum Gilbert-Varshamov (GV) bound states that there exist q-ary quantum codes of sufficiently long block length N having fixed rate R with distance at least NH_{q^2}^{-1}((1−R)/2), where H_{q^2} is the {q^2}-ary entropy function. For q<7, only random quantum codes are known to asymptotically attain the quantum GV bound. However, random codes have little structure. In this paper, we generalize the classical result of Thommesen to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region.

### 1. Truncated channel representations for coupled harmonic oscillators, Journal of Physics A: Mathematical and Theoretical

### Yingkai Ouyang, W.H. Ng

Coupled quantum harmonic oscillators, studied by many authors using many different techniques over the decades, are frequently used toy-models to study open quantum systems. In this manuscript, we explicitly study the simplest oscillator model -- a pair of initially decoupled quantum harmonic oscillators interacting with a spring-like coupling, where the bath oscillator is initially in a thermal-like state. In particular, we treat the completely positive and trace preserving map on the system as a quantum channel, and study the truncation of the channel by truncating its Kraus set and its output dimension. We thereby derive the truncated transition amplitudes of the corresponding truncated channel. Finally, we give a computable approximation for these truncated transition amplitudes with explicit error bounds, and perform a case study of the oscillators in the off-resonant and weakly-coupled regime numerically. We demonstrate explicitly that the substantial leakage error can be mitigated via quantum error correction.

### CH De Groot, Y Ouyang, Sang-Mo Koo, E Kendall, QQ Shu, JS Moodera

Tunnelling characteristics have been measured for Co-insulator-Py magnetic junctions with Dy-or Gd-doped Al 2 O 3 barriers. The theoretically predicted enhancement of the magneto-resistance due to resonant tunnelling in the presence of paramagnetic impurities is not borne out. However, even a full monolayer of Dy or Gd has no detrimental effect on the junction magneto-resistance (JMR) at low temperature and low bias voltage. With increasing temperature and bias, the JMR of the doped junctions decreases significantly faster than the JMR of the Al 2 O 3 control junctions. Junctions in which the entire barrier has been replaced by Dy 2 O 3 or Gd 2 O 3 show strong non-linear current-voltage characteristics, but display no JMR. It is shown that not direct tunnelling but spin-independent impurity-assisted tunnelling is the primary conductance channel in these junctions.